Riemannian Warped Product Maps

Abstract

In this paper, we introduce Riemannian warped product map as a generalization of warped product isometric immersion and Riemannian warped product submersion with examples and obtain some characterizations. First, we define Riemannian warped product map and find conditions for a Riemannian map to be Riemannian warped product map. We show that Riemannian warped product map is a composition of Riemannian warped product submersion followed by warped product isometric immersion locally. In addition, we show that Riemannian warped product map satisfies the generalized eikonal equation which is a well known bridge between geometrical and physical optics. We also find necessary and sufficient conditions for the fibers, range space of the derivative map of Riemannian warped product map and horizontal distributions to be totally geodesic and minimal. Further, we give some fundamental geometric properties for the study of such smooth maps. Precisely, we construct Gauss formula (second fundamental form), Weingarten formula and tension field. We obtain necessary and sufficient conditions for a Riemannian warped product map to be totally geodesic, harmonic and umbilical. Comparatively, we analyse the obtained results with the existing results for a Riemannian map between Riemannian manifolds.

1 Introduction

The notion of warped product manifolds is a generalization of the Riemannian product manifolds [3]. In general relativity, warped product manifolds have been used to construct Schwarzschild and Robertson–Walker cosmological models [24]. In addition, warped product manifolds were used to obtain new families of Hamiltonian-stationary Lagrangian submanifolds [11]. Moore gave sufficient conditions for an isometric immersion into Euclidean space to decompose into a product immersion [19]. Nash concluded that every Riemannian manifold hence warped product manifold can be embedded to some Euclidean space [20]. For the differential geometry of submanifolds of warped product manifolds, we refer to [6, 12, 13].

Let $\varphi _i: M_i \rightarrow N_i$ be isometric immersions between Riemannian manifolds for $i =1, 2, \dots , k$. Suppose $\rho _i: N_i \rightarrow {\mathbb {R}}^{+}$ and $f_i:= \rho _i \circ \varphi _i: M_i \rightarrow {\mathbb {R}}^{+}$ for $i =1, 2, \dots , k-1$ be smooth functions. Then the smooth map $\varphi : M_1 \times _{f_1} M_2 \times _{f_2} \dots \times _{f_{k-1}} M_k \rightarrow N_1 \times _{\rho _1} N_2 \times _{\rho _2} \dots \times _{\rho _{k-1}} N_k$ between warped product manifolds such that $\varphi (p_1, p_2, \dots , p_k) = (\varphi _1(p_1), \varphi _2(p_2), \dots , \varphi _k(p_k))$ is also an isometric immersion, called warped product isometric immersion. For the fundamental studies of warped product immersions we refer to [5,6,7,8,9,10, 22, 31].

The geometry of Riemannian submersions and their applications have been discussed widely by Falcitelli et al. [16]. It is known that Riemannian submersions have been used to construct some Riemannian manifolds with positive or non-negative sectional curvature and Einstein manifolds. In addition, Riemannian submersions have many applications in physics, Yang-Mills theory, Kaluza-Klein theory, supergravity and superstring theories etc. Recently, warped product Riemannian submersions were studied by Erken et al. [14, 15]. Let $\varphi _i: M_i \rightarrow N_i$ be Riemannian submersions between Riemannian manifolds for $i =1, 2, \dots , k$. Suppose $\rho _i: N_i \rightarrow {\mathbb {R}}^{+}$ and $f_i:= \rho _i \circ \varphi _i: M_i \rightarrow {\mathbb {R}}^{+}$ for $i =1, 2, \dots , k-1$ be smooth functions. Then the smooth map $\varphi : M_1 \times _{f_1} M_2 \times _{f_2} \dots \times _{f_{k-1}} M_k \rightarrow N_1 \times _{\rho _1} N_2 \times _{\rho _2} \dots \times _{\rho _{k-1}} N_k$ between warped product manifolds such that $\varphi (p_1, p_2, \dots , p_k) = (\varphi _1(p_1), \varphi _2(p_2), \dots , \varphi _k(p_k))$ is also a Riemannian submersion, called Riemannian warped product submersion.

In 1992, a generalization of the notions of an isometric immersion and Riemannian submersion namely “Riemannian map” was introduced by Fischer [17] satisfying the generalized eikonal equation whose applications are well studied in geometry and physics. Importantly, Fischer proposed an approach to build a quantum model of nature using Riemannian maps. He pointed out an interesting relationship between Riemannian maps, harmonic maps and Lagrangian field theory on the mathematical side, and Maxwell’s equation, Shrödinger’s equation and their proposed generalization on the physical side. In the last decade, Şahin investigated the geometry of Riemannian maps widely [29].

Now, it will be interesting to introduce the new area of research namely “Riemannian warped product map” as a generalization of warped product isometric immersion and Riemannian warped product submersion. In Sect. 2, we recall some basic geometric concepts about the Riemannian maps and Riemannian warped product manifolds. In Sect. 3, we define Riemannian warped product map with examples and obtain some characterizations. We show that Riemannian warped product map satisfies the generalized eikonal equation. We also find necessary and sufficient conditions for the fibers, range space of the derivative map of Riemannian warped product map and horizontal distributions to be totally geodesic and minimal. In Sect. 4, we calculate the second fundamental form (Gauss formula) followed by a necessary and sufficient condition for a Riemannian warped product map to be totally geodesic. In Sect. 5, we calculate tension field followed by a necessary and sufficient condition for a Riemannian warped product map to be harmonic. In the last section, we construct Weingarten formula followed by a necessary and sufficient condition for a Riemannian warped product map to be umbilical.

2 Preliminaries

In this section, we survey the notion of Riemannian map between Riemannian manifolds and its fundamental geometric properties. Later on we recall the notion of Riemannian warped product manifolds.

2.1 Riemannian maps

Let $\varphi :(M^m,g_M)\rightarrow (N^n,g_N)$ be a smooth map between Riemannian manifolds such that $0 < rank \varphi \le \min \{m,n\}$ and let $\varphi _{*}$ be its differential map. We denote the kernel space of $\varphi _*$ at $p\in M$ by ${{\mathcal {V}}}_p= ker\varphi _{*p}$ and its orthogonal complement space in the tangent space $T_pM$ by ${\mathcal {H}}_p=(ker\varphi _{*p})^\perp$. Thus we have

$$\begin{aligned}T_p M = (ker\varphi _{*p}) \oplus (ker\varphi _{*p})^\perp = {{\mathcal {V}}}_p \oplus {\mathcal {H}}_p.\end{aligned}$$

Similarly, we denote the range space of $\varphi _*$ at $\varphi (p) \in N$ by $range\varphi _{*p}$ and its orthogonal complement space in the tangent space $T_{\varphi (p)}N$ by $(range\varphi _{*p})^\perp$. If $rank\varphi < \min \{m,n \}$, we have $(range\varphi _*)^\perp \ne \{0\}$ and hence

$$\begin{aligned}T_{\varphi (p)}N= (range\varphi _{*p}) \oplus (range\varphi _{*p})^\perp .\end{aligned}$$

We say $\varphi$ is a Riemannian map at $p\in M$ if $\varphi _{*p}|_{{\mathcal {H}}}: ((ker\varphi _{*p})^\perp ,g_{M(p)}|_{(ker\varphi _{*p})^\perp } ) \rightarrow (range\varphi _{*p},g_{N(\varphi (p))}|_{(range\varphi _{*p})})$ is a linear isometry, i.e.

$$\begin{aligned} g_N(\varphi _*X, \varphi _*Y) =g_M(X,Y)~\hbox { for all}\ X, Y \in \Gamma (ker\varphi _{*p})^\perp . \end{aligned}$$

(1)

Clearly, it is an isometric immersion if $ker\varphi _*= \{0\}$ and a Riemannian submersion if $(range\varphi _*)^\perp = \{0\}$.

For all vector fields X, Y on M, the O’Neill tensors A and T were defined in [23]

$$\begin{aligned} A_X Y= & {} {\mathcal {H}} \nabla _{{\mathcal {H}}X}^M {{\mathcal {V}}} Y + {{\mathcal {V}}} \nabla _{{\mathcal {H}}X}^M {\mathcal {H}} Y, \end{aligned}$$

(2)

$$\begin{aligned} T_X Y= & {} {\mathcal {H}} \nabla _{{{\mathcal {V}}} X}^M {{\mathcal {V}}} Y + {{\mathcal {V}}} \nabla _{{{\mathcal {V}}} X}^M {\mathcal {H}} Y, \end{aligned}$$

(3)

where $\nabla ^M$ is the Levi-Civita connection of $g_M$. Here ${{\mathcal {V}}}$ and ${\mathcal {H}}$ denote the projections to vertical and horizontal subbundles, respectively. For any $X \in \Gamma (TM)$ the operators $T_X$ and $A_X$ are skew-symmetric reversing the horizontal and vertical distributions. In addition, $T_X = T_{{{\mathcal {V}}} X}, A_X = A_{{\mathcal {H}}X}$ and $T_U W = T_W U$ for all $U, W \in \Gamma (ker\varphi _*)$.

Now, from (2) and (3), we have

$$\begin{aligned} \nabla _V^M W= & {} T_V W + {\hat{\nabla }}_V W, \\ \nabla _X^M V= & {} A_X V + {{\mathcal {V}}} \nabla _X^M V \end{aligned}$$

and

$$\begin{aligned} \nabla _X^M Y = A_X Y + {\mathcal {H}} \nabla _X^M Y \end{aligned}$$

for all $X,Y \in \Gamma (ker\varphi _*)^\perp$ and $V,W \in \Gamma (ker\varphi _*)$ with ${\hat{\nabla }}_V W= {{\mathcal {V}}} \nabla _V^M W$.

The map $\varphi _*$ can be viewed as a section of bundle $Hom(TM,\varphi ^{-1}TN)$ $\rightarrow M$, where $\varphi ^{-1}TN$ is the pullback bundle whose fibers at $p\in M$ is $(\varphi ^{-1}TN)_p = T_{\varphi (p)}N$. The bundle $Hom(TM,\varphi ^{-1}TN)$ has a connection $\nabla$ induced from the Levi-Civita connection ${\nabla }^M$ and the pullback connection $\overset{N}{\nabla ^\varphi }$. Then the second fundamental form of $\varphi$ is given by [21]

$$\begin{aligned} (\nabla \varphi _*) (X, Y) = (\nabla \varphi _*) (Y, X)= \overset{N}{\nabla _X^\varphi }\ \varphi _*Y - \varphi _*({\nabla }_X^M Y) \end{aligned}$$

(4)

for all $X,Y \in \Gamma (TM)$, where $\overset{N}{\nabla _{X}^\varphi } \varphi _*Y \circ \varphi = \nabla _{\varphi _*X}^N \varphi _*Y$.

2.2 Riemannian warped product manifolds

Let $(M_1^{m_1},g_{M_1})$ and $(M_2^{m_2},g_{M_2})$ be two Riemannian manifolds and f be a positive smooth function on $M_1$. The warped product $M:= M_1 \times _f M_2$ of $M_1$ and $M_2$ is the Cartesian product $M_1 \times M_2$ with the metric $g_M = g_{M_1} + f^2 g_{M_2}$ defined by

$$\begin{aligned} g_M(X,Y) = g_{M_1} (\pi _{ *}(X), \pi _{*}(Y)) + f^2 (\pi (p_1))g_{M_2}(\sigma _{ *}(X), \sigma _{ *}(Y)), \end{aligned}$$

where X, Y are vector fields on $M_1 \times M_2$. In addition, $\pi : M_1 \times _f M_2 \rightarrow M_1$ such that $(x,y) \rightarrow x$ and $\sigma : M_1 \times _f M_2 \rightarrow M_2$ such that $(x,y) \rightarrow y$ are the projection maps which become submersions. Moreover, we can see that the fibers $\{x\} \times M_2 = \pi ^{-1}(x)$ and the leaves $M_1 \times \{y\} = \sigma ^{-1}(y)$ are Riemannian submanifolds of $M= M_1 \times _f M_2$. The vectors tangent to the leaves are called horizontal and the vectors tangent to the fibers are called vertical. If $v \in T_x M_1, x \in M_1$ and $y\in M_2$, then the lift ${\bar{v}}$ of v to (x, y) is the unique vector of $T_{(x,y)}M_1 \times M_2= T_{(x,y)}M$ such that $\pi _*({\bar{v}})= v$, and lift of a vector field $X \in \Gamma (TM_1)$ to $M = M_1 \times _f M_2$ is the vector field ${\bar{X}}$ such that $\pi _{*(x, y)} ({\bar{X}}_{(x, y)}) = X_x$. Thus the lift of $X\in \Gamma (TM_1)$ to $M_1 \times M_2$ is the unique element ${\bar{X}}$ of $\Gamma (T(M_1 \times M_2))$ which is $\pi$-related to X. Now, the set of all such horizontal lifts ${\bar{X}}$ is denoted by ${\mathcal {L}}_{{\mathcal {H}}}(M_1)$ and the set of all vertical lifts by ${\mathcal {L}}_{{\mathcal {V}}}(M_2)$ (for details we refer to [23] and [6]). Thus a vector field ${\bar{E}}$ of $M_1 \times M_2$ can be written as ${\bar{E}} = {\bar{X}} + {\bar{U}}$, where ${\bar{X}} \in {\mathcal {L}}_{{\mathcal {H}}}(M_1)$ and ${\bar{U}} \in {\mathcal {L}}_{{\mathcal {V}}}(M_2)$. We can easily prove that $\pi _*({\mathcal {L}}_{{\mathcal {H}}}(M_1)) = \Gamma (TM_1)$ and $\sigma _{ *}({\mathcal {L}}_{{\mathcal {V}}}(M_2)) = \Gamma (TM_2)$. Clearly, $\pi _*({\bar{X}}) = X \in \Gamma (TM_1)$ and $\sigma _{ *}({\bar{U}}) = U \in \Gamma (TM_2)$. In this paper, we use the same notation for a vector field and for its lift to the product manifold.

Now, we recall some basic results on warped product manifolds:

Lemma 1

(Chapter  7,  Lemma  34, [24, p.  206]). If ${\tilde{f}} \in {\mathcal {F}}(M_1)$, then the gradient of the lift ${\tilde{f}} \circ \pi$ of ${\tilde{f}}$ to $M = M_1 \times _f M_2$ is the lift of the gradient of ${\tilde{f}}$ on $M_1$.

Lemma 2

[24]. Let $M = M_1 \times _f M_2$ be a Riemannian warped product manifold. Then

(i):

$\nabla _{X_1}^{M} Y_1$ is the lift of $\nabla _{X_1}^{M_1} Y_1$,

(ii):

$\nabla _{X_1}^{M} X_2 = \nabla _{X_2}^{M} X_1 = (X_1(f)/f)X_2$,

(iii):

nor $(\nabla _{X_2}^{M} Y_2) = - g_M (X_2, Y_2) (\nabla ^{M} \ln f)$,

(iv):

tan $(\nabla _{X_2}^{M} Y_2)$ is the lift of $(\nabla _{X_2}^{M_2} Y_2)$,

where $X_i, Y_i \in {\mathcal {L}}(M_i)$. In addition, $\nabla ^{M}$ and $\nabla ^{M_i}$ are Levi-Civita connections on M and $M_i$ respectively for $i = 1, 2$.

Proposition 3

[4]. Let $\varphi _i: M_i \rightarrow N_i$ for $i=1,2$ be smooth functions. Then

$$\begin{aligned} (\varphi _1 \times \varphi _2)_{*} x = (\varphi _{1 *} x_1, \varphi _{2 *} x_2 ), \end{aligned}$$

where $x = (x_1, x_2) \in T_{(p_1, p_2)} (M_1 \times M_2)$.

Proposition 4

[4]. Let $\pi$ and $\sigma$ be the projections of $M_1 \times M_2$ onto $M_1$ and $M_2$ respectively. Then $\lambda : T_{(p_1, p_2)} (M_1 \times M_2) \rightarrow T_{p_1} M_1 \oplus T_{p_2} M_2$ such that $x \mapsto (\pi _*, \sigma _*) x$ is an isomorphism.

By above Proposition for $X = (X_1, X_2) \in \Gamma (T(M_1 \times M_2))$ we can write $X = X_1 + X_2$ where $X_1 \in {\mathcal {L}}(M_1)$ and $X_2 \in {\mathcal {L}}(M_2)$.

3 Characterizations of Riemannian Warped Product Maps

In this section, we introduce Riemannian warped product map between Riemannian warped product manifolds with examples and obtain some characterizations.

Proposition 5

Let $\varphi _i: (M_i, g_{M_i}) \rightarrow (N_i, g_{N_i})$ be Riemannian maps between Riemannian manifolds for $i=1,2$. Suppose $\rho : N_1 \rightarrow {\mathbb {R}}^{+}$ and $f:= \rho \circ \varphi _1: M_1 \rightarrow {\mathbb {R}}^{+}$ be smooth functions. Then the map $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ between Riemannian warped product manifolds such that $(\varphi _1 \times \varphi _2)(p_1, p_2) = (\varphi _1(p_1), \varphi _2(p_2))$ is a Riemannian map. Here f is called lift of $\rho$.

Proof

Let $\varphi _i: M_i \rightarrow N_i$ be two Riemannian maps between Riemannian manifolds for $i=1, 2$. For $p = (p_1, p_2) \in M_1 \times M_2$ we have

$$\begin{aligned}T_{(p_1, p_2)}(M_1 \times M_2)= & {} T_{(p_1, p_2)}(M_1 \times \{p_2\}) \oplus T_{(p_1, p_2)}(\{p_1\} \times M_2)\\= & {} (ker\varphi _{*p})^\perp \oplus (ker\varphi _{*p}),\end{aligned}$$

where

$$\begin{aligned}T_{(p_1, p_2)}(M_1 \times \{p_2\}) = \left( (ker\varphi _{1*p_1})^\perp \times \{p_2\} \right) \oplus \left( (ker\varphi _{1*p_1}) \times \{p_2\} \right) \end{aligned}$$

and

$$\begin{aligned}T_{(p_1, p_2)}(\{p_1\} \times M_2)= \left( \{p_1\} \times (ker\varphi _{2*p_2})^\perp \right) \oplus \left( \{p_1\} \times (ker\varphi _{2*p_2}) \right) .\end{aligned}$$

Clearly,

$$\begin{aligned}(ker\varphi _{*p})^\perp = \left( (ker\varphi _{1*p_1})^\perp \times \{p_2\} \right) \oplus \left( \{p_1\} \times (ker\varphi _{2*p_2})^\perp \right) \end{aligned}$$

and

$$\begin{aligned}(ker\varphi _{*p}) = \left( (ker\varphi _{1*p_1}) \times \{p_2\} \right) \oplus \left( \{p_1\} \times (ker\varphi _{2*p_2}) \right) .\end{aligned}$$

Similarly, we have

$$\begin{aligned} T_{({\varphi _1(p_1)}, {\varphi _2(p_2)})}(N_1 \times N_2) =&T_{({\varphi _1(p_1)}, {\varphi _2(p_2)})}(N_1 \times \{{\varphi _2(p_2)}\}) \\&\oplus T_{({\varphi _1(p_1)}, {\varphi _2(p_2)})}(\{{\varphi _1(p_1)}\} \times N_2) \\=&(range\varphi _{*p})^\perp \oplus (range\varphi _{*p}), \end{aligned}$$

where

$$\begin{aligned}{} & {} T_{({\varphi _1(p_1)}, {\varphi _2(p_2)})} (N_1 \times \{{\varphi _2(p_2)}\}) \\{} & {} \quad = \left( (range\varphi _{1*{p_1}})^\perp \times \{{\varphi _2(p_2)}\} \right) \oplus \left( (range\varphi _{1*{p_1}}) \times \{{\varphi _2(p_2)}\} \right) \end{aligned}$$

and

$$\begin{aligned}{} & {} T_{({\varphi _1(p_1)}, {\varphi _2(p_2)})}(\{{\varphi _1(p_1)}\} \times N_2)\\{} & {} \quad = \left( \{{\varphi _1(p_1)}\} \times (range\varphi _{2*{p_2}})^\perp \right) \oplus \left( \{{\varphi _1(p_1)}\} \times (range\varphi _{2*{p_2}}) \right) .\end{aligned}$$

Clearly,

$$\begin{aligned}(range\varphi _{*p})^\perp = \left( (range\varphi _{1*{p_1}})^\perp \times \{{\varphi _2(p_2)}\} \right) \oplus \left( \{{\varphi _1(p_1)}\} \times (range\varphi _{2*{p_2}})^\perp \right) \end{aligned}$$

and

$$\begin{aligned}(range\varphi _{*p}) = \left( (range\varphi _{1*{p_1}}) \times \{{\varphi _2(p_2)}\} \right) \oplus \left( \{{\varphi _1(p_1)}\} \times (range\varphi _{2*{p_2}}) \right) .\end{aligned}$$

Now, lift of a horizontal vector $x^{\mathcal {H}} \in (ker\varphi _{i *{p_i}})^\perp$ to $(ker\varphi _{*p})^\perp$ is ${\bar{x}}^{\mathcal {H}}$ and lift of a vertical tangent vector $v^{\mathcal {V}} \in (ker\varphi _{i *{p_i}})$ to $(ker\varphi _{*p})$ is ${\bar{v}}^{\mathcal {V}}$. Then for $X= (X_1, X_2), Y = (Y_1, Y_2) \in (ker\varphi _{*p})^\perp = (ker\varphi _{1*p_1})^\perp \times (ker\varphi _{2*p_2})^\perp$, we have

$$\begin{aligned} g_N(\varphi _*X, \varphi _*Y) =&g_N(\varphi _*(X_1, X_2), \varphi _*(Y_1, Y_2))\\=&g_{N_1}(\varphi _{1 *} X_1, \varphi _{1 *} Y_1) + \rho ^2(\varphi _1(p_1)) g_{N_2}(\varphi _{2 *} X_2, \varphi _{2 *} Y_2). \end{aligned}$$

Since $\varphi _1$ and $\varphi _2$ are Riemannian maps and we are denoting same notation for a vector field and its lift, using (1) in above equation, we get

$$\begin{aligned} g_N(\varphi _*X, \varphi _*Y) =&g_{M_1} (X_1, Y_1) + f^2(p_1) g_{M_2} (X_2, Y_2) \\=&g_M(X, Y). \end{aligned}$$

This implies there is an isometry between horizontal space $(ker\varphi _{*})^\perp$ and range space $range\varphi _*$. This completes the proof. $\square$

Definition 1

Let $\varphi _i: M_i \rightarrow N_i$ be Riemannian maps between Riemannian manifolds for $i=1,2$. Then the map $\varphi = \varphi _1 \times \varphi _2: (M = M_1 \times _f M_2, g_M) \rightarrow (N = N_1 \times _{\rho } N_2, g_N)$ between Riemannian warped product manifolds such that $(\varphi _1 \times \varphi _2)(p_1, p_2) = (\varphi _1(p_1), \varphi _2(p_2))$ is also a Riemannian map, called Riemannian warped product map.

Example 1

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a smooth map between Riemannian warped product manifolds.

(i):

If $\varphi$ is a warped product isometric immersion then $\varphi$ is a Riemannian warped product map with $ker \varphi _{*} = \{0\}$.

(ii):

If $\varphi$ is a Riemannian warped product submersion then $\varphi$ is a Riemannian warped product map with $(range \varphi _{*})^\perp = \{0\}$.

By [25] we know that distance functions are Riemannian submersions, hence Riemannian maps. Now, we give examples of special class of Riemannian warped product maps in the category of distance functions.

Example 2

Let $(M_i, g_{M_i})$ be Riemannian manifolds for $i= 1, 2, \dots , k$. Let $p_i \in M_i$ be a fixed point, and $d_{i}$ denote the distance functions of $M_i$ from the fixed point $p_i$. Then $\varphi :(M = M_1 \times _{f_1} M_2 \times _{f_2} \dots \times _{f_{k-1}} M_k, ~g_M) \rightarrow (N = \mathbb {R^+} \times _{\rho _1} \mathbb {R^+} \times _{\rho _2}\dots \times _{\rho _{k-1}} \mathbb {R^+}, ~g_N)$ such that

$$\begin{aligned} \varphi (q_1, q_2, \dots , q_k) = (d_1(p_1, q_1), d_2(p_2, q_2), \dots , d_k(p_k, q_k)) \end{aligned}$$

is a Riemannian warped product map, where $f_i$ is the lift of $\rho _i$ for $i = 1, 2, \dots , k-1$.

Example 3

Let $(M, g_M)$ be a complete, non-compact Riemannian manifold without conjugate points. Then every geodesic of M is a line that is it is isometric to ${\mathbb {R}}$. If $\gamma _{v}$ is a line in M, then the Busemann function $b_{v}: M \rightarrow {\mathbb {R}}$ for $\gamma _{v}$ is defined as [25]

$$\begin{aligned}b_{v}(p) = \lim \limits _{t \rightarrow \infty }^{} (d(p, \gamma _{v}(t)) - t).\end{aligned}$$

It is known that $b_v \in C^1(M)$ and $\Vert \nabla b_{v}\Vert = 1$, therefore Busemann function is a distance function. Hence clearly any Busemann function is a Riemannian map. Now let $(M_i, g_{M_i})$ be simply connected, complete manifolds without conjugate points for $i= 1, 2, \dots , k$. Then for geodesic lines $\gamma _{i}$ of $M_i$, the map $\varphi : M_1 \times _{f_1} M_2 \times _{f_2} \dots \times _{f_{k-1}} M_k \rightarrow {\mathbb {R}} \times _{\rho _1} {\mathbb {R}} \times _{\rho _2} \dots \times _{\rho _{k-1}} {\mathbb {R}}$ defined by

$$\begin{aligned} \varphi (p_1, p_2, \dots , p_k) = (b_{\gamma _{1}}(p_1), b_{\gamma _{2}}(p_2), \dots , b_{\gamma _{1}}(p_k)) \end{aligned}$$

is a Riemannian warped product map, where $f_i$ is the lift of $\rho _i$ for $i = 1, 2, \dots , k-1$.

Example 4

Let $(M_1 = \{(x_1, x_2, x_3) \in {\mathbb {R}}^3: x_1 \ne 0, x_2 \ne 0, x_3 \ne 0 \}, g_{M_1} = e^{2 x_3} d {x_1}^2 + e^{2 x_3} d {x_2}^2 + d {x_3}^2),~ (N_1 = \{(y_1, y_2, y_3) \in {\mathbb {R}}^3\}, g_{N_1} = e^{2 x_3} d {y_1}^2 + e^{2 x_3} d {y_2}^2 + d {y_3}^2),~ (M_2 = \{(r_1, r_2) \in {\mathbb {R}}^2: r_1 \ne 0, r_2 \ne 0 \}, g_{M_2} = d {r_1}^2 + d{r_2}^2)$ and $(N_2 = \{(s_1, s_2) \in {\mathbb {R}}^2 \}, g_{N_2} = d {s_1}^2 + d{s_2}^2)$ be four Riemannian manifolds. Let $\varphi _1: M_1 \rightarrow N_1$ such that

$$\begin{aligned}(x_1, x_2, x_3) \mapsto \left( \frac{x_1 + x_2}{\sqrt{2}}, \frac{x_1 - x_2}{\sqrt{2}}, 0\right) \end{aligned}$$

and $\varphi _2: M_2 \rightarrow N_2$ such that

$$\begin{aligned}(r_1, r_2) \mapsto \left( \frac{r_1 + r_2}{\sqrt{2}}, 0\right) \end{aligned}$$

be Riemannian maps. Then the map $\varphi = \varphi _1 \times \varphi _2: (M_1 \times _f M_2, g_M = g_{M_1} + f^2 g_{M_2}) \rightarrow (N_1 \times _{\rho } N_2, g_N = g_{N_1} + \rho ^2 g_{N_2})$ such that

$$\begin{aligned}\varphi (x_1, x_2, x_3, r_1, r_2) = \left( \frac{x_1 + x_2}{\sqrt{2}}, \frac{x_1 - x_2}{\sqrt{2}}, 0, \frac{r_1 + r_2}{\sqrt{2}}, 0\right) \end{aligned}$$

is a Riemannian warped product map, where $\rho : N_1 \rightarrow {\mathbb {R}}^+$ and $f: M_1 \rightarrow {\mathbb {R}}^+$ be smooth functions with $f = \rho \circ \varphi _1$.

Let $\varphi _i: M_i \rightarrow N_i$ be smooth maps between Riemannian manifolds for $i=1,2$ and let $\varphi : (M = M_1 \times M_2, g_M) \rightarrow (N = N_1 \times N_2, g_N)$ be a smooth map between Riemannian warped product manifolds. Define linear transformations

${\mathcal {P}}_{(p_1, p_2)}: T_{(p_1, p_2)}(M_1 \times M_2) \rightarrow T_{(p_1, p_2)}(M_1 \times M_2)$; ${\mathcal {P}}_{(p_1, p_2)}= {}^*\varphi _{*(p_1, p_2)} \circ \varphi _{*(p_1, p_2)}$ and ${\mathcal {Q}}_{(p_1, p_2)}: T_{(\varphi _1(p_1), \varphi _2(p_2))}(N_1 \times N_2) \rightarrow T_{(\varphi _1(p_1), \varphi _2(p_2))}(N_1 \times N_2)$; ${\mathcal {Q}}_{(p_1, p_2)}= {}^*\varphi _{*(p_1, p_2)} \circ \varphi _{*(p_1, p_2)}$ where $p_i \in M_i$ for $i = 1, 2$. In addition, ${}^*\varphi _{*}$ denotes the adjoint of $\varphi _{*}$ (see [29]). Using these linear transformations, we obtain the following characterizations of Riemannian warped product maps:

Theorem 6

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a smooth map between Riemannian warped product manifolds. Then the following statements are equivalent:

(i):

$\varphi$ is Riemannian warped product map at $(p_1, p_2) \in M$.

(ii):

${\mathcal {P}}_{(p_1, p_2)}$ is a projection, i.e. ${\mathcal {P}}_{(p_1, p_2)} \circ {\mathcal {P}}_{(p_1, p_2)} = {\mathcal {P}}_{(p_1, p_2)}$.

(iii):

${\mathcal {Q}}_{(p_1, p_2)}$ is a projection, i.e. ${\mathcal {Q}}_{(p_1, p_2)} \circ {\mathcal {Q}}_{(p_1, p_2)} = {\mathcal {Q}}_{(p_1, p_2)}$.

Proof

Since the proof is similar to Theorem 80 of [29], we are omitting it. $\square$

Now, we recall that a map $\varphi : (M, g_M) \rightarrow (N, g_N)$ between Riemannian manifolds is called subimmersion at $p \in M$ if there is an open neighborhood ${{\mathcal {U}}}$ of p, a manifold $M'$, a submersion $\varphi _S: {{\mathcal {U}}} \rightarrow M'$, and an immersion $\varphi _I: M' \rightarrow N$ such that $\varphi |_{{\mathcal {U}}} = \varphi _{{\mathcal {U}}} = \varphi _I \circ \varphi _S$ [17, 29]. We say $\varphi$ is a subimmersion if it is subimmersion at each $p \in M$. It is known that $\varphi : (M, g_M) \rightarrow (N, g_N)$ is a subimmersion if and only if the rank function is locally constant, and hence constant on the connected components of M [1, 27].

Definition 2

A map $\varphi = \varphi _1 \times \varphi _2: (M = M_1 \times _f M_2, g_M) \rightarrow (N = N_1 \times _{\rho } N_2, g_N)$ between Riemannian warped product manifolds is called warped product subimmersion at $p = (p_1, p_2) \in M_1 \times M_2$ if there is an open neighborhood ${\mathcal {U}} = {{\mathcal {U}}_1} \times _f {{\mathcal {U}}_2}$ of $p = (p_1, p_2)$, a warped product manifold $M_1' \times _{f'} M_2'$, a warped product submersion $\varphi _S: {{\mathcal {U}}_1} \times _f {{\mathcal {U}}_2} \rightarrow M_1' \times _{f'} M_2'$, and a warped product immersion $\varphi _I: M_1' \times _{f'} M_2' \rightarrow N_1 \times _{\rho } N_2$ such that $\varphi |_{{\mathcal {U}}} = \varphi _{{{\mathcal {U}}}} = \varphi _I \circ \varphi _S$, where $M_1'$ and $M_2'$ are Riemannian manifolds and $f'$ is warping function on $M_1'$. We say $\varphi$ is a warped product subimmersion if it is warped product subimmersion at each $p \in M_1 \times M_2$.

Theorem 7

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then, locally, $\varphi$ is the composition of a Riemannian warped product submersion followed by a warped product isometric immersion.

Proof

Let ${\mathcal {U}} = {\mathcal {U}}_1 \times _f {\mathcal {U}}_2$ be an open neighborhood of $p = (p_1, p_2) \in M_1 \times M_2$, $\varphi _S: {{\mathcal {U}}_1} \times _f {{\mathcal {U}}_2} \rightarrow M_1' \times _{f'} M_2'$ be a warped product submersion and $\varphi _I: M_1' \times _{f'} M_2' \rightarrow N_1 \times _{\rho } N_2$ be a warped product immersion such that $\varphi |_{{\mathcal {U}}} = \varphi _{{{\mathcal {U}}}} = \varphi _I \circ \varphi _S$. From Lemma 4.3.1 of [18], it follows that $(M_1' \times _{f'} M_2', g_{M'})$ is also a Riemannian warped product manifold and $\varphi _I: (M'_1 \times _{f'} M'_2, g_M') \rightarrow (N_1 \times _\rho N_2, g_N)$ is a warped product isometric immersion. Therefore it is enough to show that horizontal restriction of $\varphi _S$ is a warped product submersion. As in [27], for each $p = (p_1, p_2) \in {\mathcal {U}}_1 \times {\mathcal {U}}_2$ define

$$\begin{aligned}{\varphi _S}_{*p}|_{{\mathcal {H}}}: (ker \varphi _{1 *p_1})^\perp \times (ker \varphi _{2 *p_2})^\perp \rightarrow T_{\varphi _{S(p)}} (M'_1 \times M'_2)\end{aligned}$$

as ${\varphi _S}_{*p}|_{{\mathcal {H}}} X= {\varphi _S}_{*p} X$ and

$$\begin{aligned}{\varphi _I}_{*p}|_{{\mathcal {H}}}: T_{\varphi _{S(p)}} (M'_1 \times M'_2) \rightarrow ({range \varphi _{1*p_1}}) \times ({range \varphi _{2 *p_2}})\end{aligned}$$

as ${\varphi _I}_{*p}|_{{\mathcal {H}}} Z= {\varphi _I}_{*p} Z$. Thus we have $\varphi _{*p}|_{{\mathcal {H}}} = ({\varphi _I}_{*p}|_{{\mathcal {H}}} \circ {\varphi _S}_{*p}|_{{\mathcal {H}}}) = ({\varphi _I}_{*p} \circ {\varphi _S}_{*p})|_{{\mathcal {H}}}: ker \varphi _{*p} = (ker \varphi _{ 1 *p_1})^\perp \times (ker \varphi _{2 *p_2})^\perp \rightarrow range \varphi _{*p} = ({range \varphi _{1*p_1}}) \times ({range \varphi _{2 *p_2}})$. Then for $X = (X_1, X_2)$ and $Y = (Y_1, Y_2) \in (ker \varphi _{1 *p_1})^\perp \times (ker \varphi _{2 *p_2})^\perp$, we get

$$\begin{aligned}g_N(\varphi _{*p}|_{{\mathcal {H}}} X, \varphi _{*p}|_{{\mathcal {H}}} Y) = g_N({\varphi _I}_{*p}|_{{\mathcal {H}}} \circ {\varphi _S}_{*p}|_{{\mathcal {H}}} X, {\varphi _I}_{*p}|_{{\mathcal {H}}} \circ {\varphi _S}_{*p}|_{{\mathcal {H}}} Y).\end{aligned}$$

Since $\varphi : (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ is a Riemannian warped product map and $\varphi _I: (M'_1 \times _{f'} M'_2, g_M') \rightarrow (N_1 \times _\rho N_2, g_N)$ is a warped product isometric immersion, we get $g_{M'} = \varphi _I^{*} g_N$. Then

$$\begin{aligned}&g_N(\varphi _{*p}|_{{\mathcal {H}}} (X_1, X_2), \varphi _{*p}|_{{\mathcal {H}}} (Y_1, Y_2))\\&= g_N({\varphi _I}_{*p}|_{{\mathcal {H}}} \circ {\varphi _S}_{*p}|_{{\mathcal {H}}} (X_1, X_2), {\varphi _I}_{*p}|_{{\mathcal {H}}} \circ {\varphi _S}_{*p}|_{{\mathcal {H}}} (Y_1, Y_2)), \end{aligned}$$

which implies

$$\begin{aligned} g_M|_{\mathcal {U}} ((X_1, X_2), (Y_1, Y_2))= g_{M'}({\varphi _S}_{*p}|_{{\mathcal {H}}} (X_1, X_2), {\varphi _S}_{*p}|_{{\mathcal {H}}} (Y_1, Y_2)). \end{aligned}$$

Thus $\varphi _S: ({{\mathcal {U}}_1} \times _f {{\mathcal {U}}_2}, g_M|_{\mathcal {U}}) \rightarrow (M_1' \times _{f'} M_2', g_{M'})$ is a Riemannian warped product submersion. Hence we finish the required proof. $\square$

Remark 1

Note that a Riemannian map between Riemannian manifolds is a composition of a Riemannian submersion followed by an isometric immersion [17] locally. It is also true that a Riemannian warped product map is a composition of a Riemannian warped product submersion followed by warped product isometric immersion locally.

Now, we give the following examples of Riemannian warped product maps as an application of the above Theorem 7:

Example 5

Consider the immersions $\varphi _1: {\mathbb {R}}^{+} \rightarrow {\mathbb {E}}^2=\{(x, y) \in {\mathbb {R}}^{2}: y > 0\}$ such that

$$\begin{aligned}\varphi _1 (t) = (t, t)\end{aligned}$$

and $\varphi _2: {\mathbb {S}}^{1} \rightarrow {\mathbb {S}}^1$ such that

$$\begin{aligned}\varphi _2 (s) = s.\end{aligned}$$

Then by [22], $\varphi = \varphi _1 \times \varphi _2: {\mathbb {R}}^{+} \times _{f} {\mathbb {S}}^{1} \rightarrow {\mathbb {E}}^2 \times _{f'} {\mathbb {S}}^{1}$ defined by

$$\begin{aligned}\varphi (t, s) = (t, t, s)\end{aligned}$$

is a warped product isometric immersion with $(f'\circ \varphi _1 )(t) = t = f(t)$. Also, consider the Riemannian submersions $\psi _1: {\mathbb {F}}^2 = \{(t_1, t_2) \in {\mathbb {R}}^{2}: t_1 > 0\} \rightarrow {\mathbb {R}}^{+}$ such that

$$\begin{aligned}\psi _1 (t_1, t_2) = t_1\end{aligned}$$

and $\psi _2: {\mathbb {S}}^1 \rightarrow {\mathbb {S}}^{1}$ such that

$$\begin{aligned}\psi _2 (s) = s.\end{aligned}$$

Then by [14], $\psi = \psi _1 \times \psi _2: {\mathbb {F}}^2 \times _{f'} {\mathbb {S}}^1 \rightarrow {\mathbb {R}}^{+} \times _\rho {\mathbb {S}}^{1}$ defined by

$$\begin{aligned} \psi (t_1, t_2, s) = (t_1, s) \end{aligned}$$

is a Riemannian warped product submersion with $(\rho \circ \psi _1) (t_1, t_2) = t_1 = f'(t_1,t_2)$. Thus $\varphi \circ \psi : {\mathbb {R}}^2 \times _{f'} {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2 \times _{f'} {\mathbb {R}}^2$ is a Riemannian warped product map.

Example 6

Consider the immersions $\varphi _i: {\mathbb {R}} \rightarrow {\mathbb {R}}^2$ for $i = 1, 2$ such that

$$\begin{aligned}\varphi _i (t) = (\cos t, \sin t).\end{aligned}$$

Then by [22], $\varphi = \varphi _1 \times \varphi _2: {\mathbb {R}} \times _{f} {\mathbb {R}} \rightarrow {\mathbb {R}}^2 \times _{f'} {\mathbb {R}}^2$ defined by

$$\begin{aligned}\varphi (t_1, t_2) = (\cos t_1, \sin t_1, \cos t_2, \sin t_2)\end{aligned}$$

is a warped product isometric immersion with $f' \circ \varphi _1 = f$. Also, consider the Riemannian submersions $\psi _i: {\mathbb {R}}^2 \rightarrow {\mathbb {R}}$ for $i = 1, 2$ such that

$$\begin{aligned}\psi _i (x_1, x_2) = x_1.\end{aligned}$$

Then by [14], $\psi = \psi _1 \times \psi _2: {\mathbb {R}}^2 \times _{f'} {\mathbb {R}}^2 \rightarrow {\mathbb {R}} \times _\rho {\mathbb {R}}$ defined by

$$\begin{aligned}\psi (x_1, x_2, y_1, y_2) = (x_1, y_1)\end{aligned}$$

is a Riemannian warped product submersion with $\rho \circ \psi _1 = f'$. Thus $\varphi \circ \psi : {\mathbb {R}}^2 \times _{f'} {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^2 \times _{f'} {\mathbb {R}}^2$ is a Riemannian warped product map.

Theorem 8

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then

$$\begin{aligned} \Vert \varphi _*\Vert ^2 = rank \varphi _1 + \rho ^2 rank \varphi _2. \end{aligned}$$

Proof

Define a linear transformation $G: ((ker\varphi _{*p})^\perp = (ker\varphi _{1 *p_1})^\perp \times (ker\varphi _{2 *p_2})^\perp , g_{M p})$ $\rightarrow ((ker\varphi _{*p})^\perp = (ker\varphi _{1 *p_1})^\perp \times (ker\varphi _{2 *p_2})^\perp , g_{M p})$ such that $G=$ $^{*}\varphi _{*p} \circ \varphi _{*p}$, where $^{*}\varphi _{*p}$ is the adjoint of $\varphi _{*p}$. Then for $X= (X_1, X_2), Y = (Y_1, Y_2) \in (ker\varphi _{*p})^\perp$, we have

$$\begin{aligned} g_M(G X, Y) =&g_M(^{*}\varphi _{*p} \circ \varphi _{*p} X, Y) = g_N(\varphi _{*p}X, \varphi _{*p}Y) \\ =&g_{N_1}(\varphi _{1 *}X_1, \varphi _{1 *} X_2) + \rho ^2(\varphi _1(p_1)) g_{N_2}(\varphi _{2 *}Y_1, \varphi _{2 *} Y_2). \end{aligned}$$

Since $\varphi _1$ and $\varphi _2$ are Riemannian maps, using (1) in above equation, we get

$$\begin{aligned} g_M(G X, Y) = g_{M_1}(X_1, X_2) + f^2(p_1) g_{M_2}(Y_1, Y_2). \end{aligned}$$

Now let $\{{{\tilde{x}}_i}\}_{i=1}^{m_1+m_2-r_1-r_2}, \{x_j\}_{j=r_1 +1}^{m_1}$ and $\{x_k^*\}_{k=m_1-r_1+1}^{m_1 - r_1 + m_2 -r_2}$ denote orthonormal bases of $(ker\varphi _{*p})^\perp$, $(ker\varphi _{1 *p_1})^\perp$, and $(ker\varphi _{2 *p_2})^\perp$, respectively, and ${{\tilde{x}}_i} = (x_j, x_k^*) \in (ker\varphi _{*p})^\perp$. Then we have

$$\begin{aligned} \begin{array}{llll} \Vert \varphi _*\Vert ^2 (p) &{}= \sum \limits _{i=1}^{m_1+m_2-r_1-r_2} g_N(\varphi _{*p}{{\tilde{x}}_i}, \varphi _{*p}{{\tilde{x}}_i})\\ {} &{}= \sum \limits _{j=1 + r_1}^{m_1} g_{N_1}(\varphi _{1 *}x_j, \varphi _{1 *}x_j) \\ {} &{}+ \rho ^2(\varphi _1(p_1)) \sum \limits _{k = m_1 -r_1 +1}^{m_2-r_2+m_1-r_1} g_{N_2}(\varphi _{2 *} x_k^*, \varphi _{2 *}x_k^*)\\ &{}= (m_1 - r_1) + \rho ^2(\varphi _1(p_1)) (m_2 - r_2), \end{array} \end{aligned}$$

where $m_1 + m_2 -r_1 -r_2 = \dim (ker\varphi _*)^\perp , r_1 = \dim (ker\varphi _{1 *}), r_2 = \dim (ker\varphi _{2 *}), m_1 = \dim (M_1)$ and $m_2 = \dim (M_2)$. This completes the proof. $\square$

Remark 2

We can observe that a Riemannian map $\varphi$ between Riemannian manifolds satisfies $\Vert \varphi _*\Vert ^2 = rank \varphi$ [17], while a Riemannian warped product map $\varphi = \varphi _1 \times \varphi _2: M_1 \times _f M_2 \rightarrow N_1 \times _{\rho } M_2$ between Riemannian warped product manifolds satisfies $\Vert \varphi _*\Vert ^2 = rank \varphi _1 + \rho ^2 rank \varphi _2$. Hence Riemannian warped product map satisfies the generalized eikonal equation.

Lemma 9

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a smooth map between Riemannian warped product manifolds. Then

(i):

$\nabla _{\varphi _{1*} X_1}^{N} \varphi _{1 *}Y_1$ is the lift of $\nabla _{\varphi _{1*}X_1}^{N_1} \varphi _{1 *}Y_1$,

(ii):

$\nabla _{\varphi _{1*} X_1}^{N} \varphi _{2 *}X_2$= $\nabla _{\varphi _{2*}X_2}^{N} \varphi _{1 *}X_1$= $(\varphi _{1 *}X_1(\rho )/\rho )\varphi _{2 *}X_2$,

(iii):

tan $\nabla _{\varphi _{2*} X_2}^{N} \varphi _{2 *}Y_2$ is the lift of $\nabla _{\varphi _{2*} X_2}^{N_2} \varphi _{2 *}Y_2$,

(iv):

nor $\nabla _{\varphi _{2*} X_2}^{N} \varphi _{2 *}Y_2$ = $-g_N(\varphi _{2 *}X_2, \varphi _{2 *}Y_2) (\nabla ^{N} \ln \rho )$,

where $\varphi _{i *}X_i, \varphi _{i *}Y_i \in {\mathcal {L}}(N_i)$. In addition, $\nabla ^{N}$ and $\nabla ^{N_i}$ are the Levi-Civita connections on N and $N_i$ respectively for $i = 1, 2$.

Proof

Since $N= N_1 \times _{\rho } N_2$ is a Riemannian warped product manifold, by using Lemma 2 we can easily see that (i) and (ii) are hold. Now for $\varphi _{1 *}X_1, \varphi _{1 *}Y_1 \in {\mathcal {L}}(N_1)$ and $\varphi _{1 *}X_2, \varphi _{1 *}Y_2 \in {\mathcal {L}}(N_2)$, we write

$$\begin{aligned} \nabla _{\varphi _{2 *}X_2}^{N} \varphi _{2 *}Y_2 = nor \nabla _{\varphi _{2 *}X_2}^{N} \varphi _{2 *}Y_2 + tan \nabla _{\varphi _{2 *}X_2}^{N} \varphi _{2 *}Y_2. \end{aligned}$$

Using Lemma 2, we get

$$\begin{aligned} \nabla _{\varphi _{2 *}X_2}^{N} \varphi _{2 *}Y_2 = -g_N(\varphi _{2 *}X_2, \varphi _{2 *}Y_2) (\nabla ^{N} \ln \rho )+ \nabla _{\varphi _{2 *}X_2}^{N_2} \varphi _{2 *}Y_2. \end{aligned}$$

Thus (iii) and (iv) also hold. $\square$

Proposition 10

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then

$$\begin{aligned}\varphi _{1 *} X_1(\ln \rho ) = X_1(\ln f) =X_1(f)/(f),\end{aligned}$$

where $X_1 \in \Gamma (ker \varphi _{1 *})^\perp$.

Proof

For $X_1 \in \Gamma (ker \varphi _{1 *})^\perp$, we have

$$\begin{aligned} \varphi _{1 *} X_1(\ln \rho ) = g_{N_1}(\varphi _{1 *} X_1, \nabla ^{N_1} (\ln \rho )). \end{aligned}$$

Since f is lift of $\rho$, using Lemma 1 in above equation, we get

$$\begin{aligned} \varphi _{1 *} X_1(\ln \rho ) = g_{N_1}(\varphi _{1 *} X_1, \varphi _{1 *} (\nabla ^{M_1} (\ln f))). \end{aligned}$$

Since $\varphi _{1}$ is a Riemannian map, we can write

$$\begin{aligned} \varphi _{1 *} X_1(\ln \rho ) = g_{M_1}(X_1, \nabla ^{M_1} (\ln f)). \end{aligned}$$

This completes the proof. $\square$

Theorem 11

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then the following statements are hold:

(i):

The distribution $ker\varphi _{*}$ is totally geodesic if and only if the distributions $ker\varphi _{1 *}$ and $ker\varphi _{2 *}$ are totally geodesic, and ${\mathcal {H}}(\nabla ^M \ln f) = 0$,

(ii):

The distribution $ker\varphi _{*}$ is minimal if and only if either the distribution $ker\varphi _{1 *}$ is minimal and $H_2 = {\mathcal {H}}(\nabla ^M \ln f)$ or the distribution $ker\varphi _{2 *}$ is minimal and $H_1 = \left( \frac{r_2}{r_1}\right) {\mathcal {H}}(\nabla ^M \ln f)$,

(iii):

The distribution $(ker\varphi _{*})^\perp$ is totally geodesic if and only if the distributions $(ker\varphi _{1*})^\perp$ and $(ker\varphi _{2*})^\perp$ are totally geodesic, and ${\mathcal {V}}(\nabla ^M \ln f) = 0$,

(iv):

The distribution $range \varphi _{*}$ is minimal if and only if either the distribution $range \varphi _{1 *}$ is minimal and $H_4 = \nabla ^N \ln \rho$ or the distribution $range \varphi _{2 *}$ is minimal and $H_3 = \left( \frac{m_2 - r_2}{m_1 - r_1}\right) \nabla ^N \ln \rho$,

(v):

The distribution $(ker \varphi _{*})^\perp$ is minimal if and only if either the distribution $(ker\varphi _{1 *})^\perp$ is minimal and $H_2^\perp = {\mathcal {V}} (\nabla ^M \ln f)$ or the distribution $(ker\varphi _{2 *})^\perp$ is minimal and $H_1^\perp = \left( \frac{m_2 - r_2}{m_1 - r_1}\right) {\mathcal {V}} (\nabla ^M \ln f)$,

(vi):

The distribution $(range \varphi _{*})^\perp$ is minimal if and only if either the distribution $(range \varphi _{1 *})^\perp$ is minimal and $H_4^\perp = \nabla ^N \ln \rho$ or the distribution $(range \varphi _{2 *})^\perp$ is minimal and $H_3^\perp = \left( \frac{n_4}{n_3}\right) \nabla ^N \ln \rho$,

where $\nabla ^M$ and $\nabla ^N$ are Levi-Civita connections on M and N respectively. In addition $H_1, H_2, H_3, H_4, H_1^\perp , H_2^\perp , H_3^\perp$ and $H_4^\perp$ are the mean curvature vector fields of $ker\varphi _{1 *}, ker\varphi _{2 *}$, $range\varphi _{1 *}$, $range\varphi _{2 *}$, $(ker\varphi _{1 *})^\perp , (ker\varphi _{2 *})^\perp$, $(range\varphi _{1 *})^\perp$ and $(range\varphi _{2 *})^\perp$ respectively. Also here $r_1 = \dim (ker\varphi _{1 *}), r_2 = \dim (ker \varphi _{2 *}), m_1-r_1 = \dim (range\varphi _{1 *})$, $m_2-r_2 = \dim (range \varphi _{2 *})$, $n_3 = \dim ((range\varphi _{1 *})^\perp )$ and $n_4 = \dim ((range\varphi _{2 *})^\perp )$.

Proof

Let $\{u_i\}_{i =1}^{r_1}$ and $\{u_a^*\}_{a = r_1 + 1}^{r_1 + r_2}$ be orthonormal bases of $ker\varphi _{1 *}$ and $ker\varphi _{2 *}$, respectively. Then

$$\begin{aligned} \Vert T\Vert ^2&= \sum \limits _{i,j = 1}^{r_1} g_M(T(u_i, u_j), T(u_i, u_j)) + \sum \limits _{a,b = r_1 + 1}^{r_1 + r_2} g_M(T(u_a^*, u_b^*), T(u_a^*, u_b^*)) \\ {}&= \sum \limits _{i,j = 1}^{r_1} g_M (T_1(u_i, u_j), T_1(u_i, u_j)) \\ {}&\quad + \sum \limits _{a,b = r_1 + 1}^{r_1 + r_2} g_M(T_2(u_a^*, u_b^*) - g_M(u_a^*, u_b^*)({\mathcal {H}} \nabla ^M \ln f), T_2(u_a^*, u_b^*) \\&\quad - g_M(u_a^*, u_b^*)({\mathcal {H}} \nabla ^M \ln f)), \end{aligned}$$

where $T, T_1$ and $T_2$ are O’Neill tensors on $ker\varphi _*, ker\varphi _{1 *}$ and $ker\varphi _{2 *}$ respectively [14]. Thus

$$\begin{aligned} \Vert T\Vert ^2 = \Vert T_1\Vert ^2 + \Vert T_2\Vert ^2 + r_2 ~\Vert {\mathcal {H}} (\nabla ^M \ln f)\Vert . \end{aligned}$$

(5)

In addition the mean curvature vector field of $ker \varphi _*$ is given by

$$\begin{aligned}&H = \frac{1}{r_1 + r_2} \left( \sum \limits _{i = 1}^{r_1} T(u_i, u_i) + \sum \limits _{a = r_1 + 1}^{r_1 + r_2} T(u_a^*, u_a^*)\right) \nonumber \\ {}&= \frac{1}{r_1 + r_2} \left( \sum \limits _{i = 1}^{r_1} T_1(u_i, u_i) + \sum \limits _{a = r_1 + 1}^{r_1 + r_2} T_2(u_a^*, u_a^*) - g_M(u_a^*, u_a^*) ({\mathcal {H}} \nabla ^M \ln f)\right) . \end{aligned}$$

(6)

We know that [29]

$$\begin{aligned} H_1 = \frac{1}{r_1} \sum _{i=1}^{r_1} T_1(u_i, u_i) \end{aligned}$$

(7)

and

$$\begin{aligned} H_2 = \frac{1}{r_2} \sum _{a= r_1 + 1}^{r_1 + r_2} T_2(u_a^*, u_a^*), \end{aligned}$$

(8)

where $H_1$ and $H_2$ denote the mean curvature vector fields of the distributions $ker\varphi _{1 *}$ and $ker\varphi _{2 *}$, respectively. Using (7) and (8) in (6), we get

$$\begin{aligned} H = \frac{1}{r_1 + r_2} \left( r_1 H_1 + r_2 (H_2 - {\mathcal {H}} \nabla ^M \ln f)\right) . \end{aligned}$$

(9)

Further, let $\{x_i\}_{i = r_1 + 1}^{m_1}$ and $\{x_a^*\}_{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2}$ be orthonormal bases of $(ker\varphi _{1 *})^\perp$ and $(ker\varphi _{2 *})^\perp$, respectively. Then

$$\begin{aligned} \Vert A\Vert ^2&= \sum \limits _{i,j = r_1 + 1}^{m_1} g_M(A(x_i, x_j), A(x_i, x_j)) + \sum \limits _{a,b = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} g_M(A(x_a^*, x_b^*), A(x_a^*, x_b^*)) \\ {}&= \sum \limits _{i,j = r_1 + 1}^{m_1} g_M (A_1(x_i, x_j), A_1(x_i, x_j)) \\ {}&\quad + \sum \limits _{a,b = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} g_M(A_2(x_a^*, x_b^*) - g_M(x_a^*, x_b^*)({\mathcal {V}} \nabla ^M \ln f), A_2(x_a^*, x_b^*) \\&\quad - g_M(x_a^*, x_b^*)({\mathcal {V}} \nabla ^M \ln f)), \end{aligned}$$

where $A, A_1$ and $A_2$ are O’Neill tensors on $(ker\varphi _*)^\perp , (ker\varphi _{1 *})^\perp$ and $(ker\varphi _{2 *})^\perp$ respectively [14]. Thus

$$\begin{aligned} \Vert A\Vert ^2 = \Vert A_1\Vert ^2 + \Vert A_2\Vert ^2 + (m_2 - r_2) ~\Vert {\mathcal {V}}(\nabla ^M \ln f)\Vert . \end{aligned}$$

(10)

In addition the mean curvature vector field of $range\varphi _{*}$ is given by

$$\begin{aligned} H' = \frac{1}{m_1 - r_1 + m_2 - r_2} \left( \sum \limits _{i = r_1 + 1}^{m_1} \nabla _{x_i}^{\varphi } \varphi _{1 *}x_i + \sum \limits _{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} \nabla _{x_a^*}^{\varphi } \varphi _{2 *}x_a^*\right) . \end{aligned}$$

(11)

Using Lemma 9 in (11), we get

$$\begin{aligned} H' =&\frac{1}{m_1 - r_1 + m_2 - r_2} \sum \limits _{i = r_1 + 1}^{m_1} \nabla _{x_i}^{\varphi _1} \varphi _{1 *}x_i \nonumber \\ {}&+ \frac{1}{m_1 - r_1 + m_2 - r_2} \sum \limits _{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} \nabla _{x_a^*}^{\varphi _2} \varphi _{2 *}x_a^* - g_M(x_a^*, x_a^*) \nabla ^N \ln \rho . \end{aligned}$$

(12)

We know that [29]

$$\begin{aligned} H_3 = \frac{1}{m_1-r_1} \sum _{i = r_1 + 1}^{m_1} \nabla _{x_i}^{\varphi _1} \varphi _{1 *}x_i \end{aligned}$$

(13)

and

$$\begin{aligned} H_4 = \frac{1}{m_2 - r_2} \sum \limits _{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} \nabla _{x_a^*}^{\varphi _2} \varphi _{2 *}x_a^*, \end{aligned}$$

(14)

where $H_3$ and $H_4$ denote the mean curvature vector fields of the distributions $range\varphi _{1 *}$ and $range\varphi _{2 *}$, respectively. Using (13) and (14) in (12), we get

$$\begin{aligned} H' = \frac{1}{m_1 - r_1 + m_2 - r_2} \left( (m_1 - r_1) H_3 + (m_2 - r_2) (H_4 - \nabla ^N \ln \rho )\right) . \end{aligned}$$

(15)

Also, the mean curvature vector field of $(ker \varphi _*)^\perp$ is given by

$$\begin{aligned} H^\perp&= \frac{1}{m_1 - r_1 + m_2 - r_2} \left( \sum \limits _{i =r_1 + 1}^{m_1} A(x_i, x_i) + \sum \limits _{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} A(x_a^*, x_a^*)\right) \nonumber \\ {}&= \frac{1}{m_1 - r_1 + m_2 - r_2} \left( \sum \limits _{i = r_1 + 1}^{m_1} A_1(x_i, x_i) + \sum \limits _{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} A_2(x_a^*, x_a^*) \right. \nonumber \\&\quad \left. - g_M(x_a^*, x_a^*) ({\mathcal {V}} \nabla ^M \ln f)\right) . \end{aligned}$$

(16)

We know that [29]

$$\begin{aligned} H_1^\perp = \frac{1}{m_1 - r_1} \sum _{i=r_1 + 1}^{m_1} A_1(x_i, x_i) \end{aligned}$$

(17)

and

$$\begin{aligned} H_2^\perp = \frac{1}{m_2 - r_2} \sum _{a= m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} A_2(x_a^*, x_a^*), \end{aligned}$$

(18)

where $H_1^\perp$ and $H_2^\perp$ denote the mean curvature vector fields of the distributions $(ker\varphi _{1 *})^\perp$ and $(ker\varphi _{2 *})^\perp$, respectively. Using (17) and (18) in (16), we get

$$\begin{aligned} H^\perp = \frac{1}{m_1 - r_1 + m_2 - r_2} \left( (m_1 -r_1) H_1^\perp + (m_2 - r_2) (H_2^\perp - {\mathcal {V}} \nabla ^M \ln f)\right) . \end{aligned}$$

(19)

Finally, let $\{\bar{e_l}\}_{l = 1}^{n_3}$ and $\{{\check{e}}_s\}_{s = n_3 + 1}^{n_3 + n_4}$ be orthonormal bases of $(range \varphi _{1 *})^\perp$ and $(range \varphi _{2 *})^\perp$, respectively. Then the mean curvature vector field of $(range\varphi _{*})^\perp$ is

$$\begin{aligned} H'^\perp = \frac{1}{n_3 + n_4} \left( \sum \limits _{l = 1}^{n_3} \nabla _{\bar{e_l}}^{\varphi ^\perp } \bar{e_l}+ \sum \limits _{s = 1 + n_3}^{n_3 + n_4} \nabla _{{\check{e}}_s}^{\varphi ^\perp } {\check{e}}_s\right) . \end{aligned}$$

(20)

Using Lemma 9 in (20), we get

$$\begin{aligned} H'^\perp = \frac{1}{n_3 + n_4} \left( \sum \limits _{l =1}^{n_3} \nabla _{\bar{e_l}}^{\varphi _1^\perp } \bar{e_l} + \sum \limits _{s = 1 + n_3}^{n_3 + n_4} \nabla _{{\check{e}}_s}^{\varphi _2^\perp } {\check{e}}_s - g_N({\check{e}}_s, {\check{e}}_s) \nabla ^N \ln \rho \right) . \end{aligned}$$

(21)

We know that [29]

$$\begin{aligned} H_3^\perp = \frac{1}{n_3} \sum _{l =1}^{n_3} \nabla _{\bar{e_l}}^{\varphi _1^\perp } \bar{e_l} \end{aligned}$$

(22)

and

$$\begin{aligned} H_4^\perp = \frac{1}{n_4} \sum \limits _{s = 1 + n_3}^{n_3 + n_4} \nabla _{{\check{e}}_s}^{\varphi _2^\perp } {\check{e}}_s , \end{aligned}$$

(23)

where $H_3^\perp$ and $H_4^\perp$ denote the mean curvature vector fields of the distributions $(range \varphi _{1 *})^\perp$ and $(range \varphi _{2 *})^\perp$, respectively. Using (22) and (23) in (21), we get

$$\begin{aligned} H'^\perp = \frac{1}{n_3 + n_4} \left( n_3 ~H_3^\perp + n_4 (H_4^\perp - \nabla ^N \ln \rho )\right) . \end{aligned}$$

(24)

Then proof follows by (5), (9), (10), (15), (19) and (24). $\square$

4 Totally Geodesic Riemannian Warped Product Maps

In this section, we construct the Gauss formula (second fundamental form) for a Riemannian warped product map between Riemannian warped product manifolds and discuss totally geodesicity [32].

Theorem 12

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then the second fundamental form of $\varphi$ is

$$\begin{aligned} (\nabla \varphi _*)(X,Y)=&(\nabla ^1 \varphi _{1 *})(X_1, Y_1) + (\nabla ^2 \varphi _{2 *})(X_2, Y_2) \nonumber \\ {}&+ (\varphi _{1 *}Y_1(\ln \rho )) \varphi _{2 *}X_2 + (\varphi _{1 *} X_1(\ln \rho )) \varphi _{2 *}Y_2 \\ {}&- (X_1(\ln f)) \varphi _{2 *} Y_2 - (Y_1(\ln f)) \varphi _{2 *}X_2,\nonumber \end{aligned}$$

(25)

where $X= (X_1, X_2),~ Y = (Y_1, Y_2) \in \Gamma (T(M_1 \times M_2))$ and $\varphi _i: M_i \rightarrow N_i$ are Riemannian maps between Riemannian manifolds for $i=1,2$. In addition, the bundle $TM_i^{*} \otimes \varphi _i^{-1}(TN_i)$ has connection $\nabla ^{i}$ induced from the Levi-Civita connection $\nabla ^{M_i}$ of $M_i$ for $i = 1, 2$. This is known as Gauss formula also.

Proof

Let $\varphi _i: M_i \rightarrow N_i$ be Riemannian maps between Riemannian manifolds and $TN_i$ be bundle over $N_i$ for $i=1, 2$. Now, the pullback bundle $\varphi _i^{-1}(TN_i) \rightarrow M_i$ has the fibers $(\varphi _i^{-1}(TN_i))_{p_i} = T_{\varphi _i(p_i)} N_i$ for $p_i \in M_i$. We shall use this identification without comment. The Levi-Civita connection $\nabla ^{N_i}$ on $N_i$ and the pullback connection $\nabla ^{\varphi _i}$ are the unique linear connections on the pullback bundle $\varphi _i^{-1}(TN_i)$ such that for each $Z_i \in \Gamma (TN_i)$

$$\begin{aligned} \nabla _{X_i}^{\varphi _i} (\varphi _i^{*} Z_i) = \nabla _{\varphi _{i *} X_i}^{N_i} Z_i, \end{aligned}$$

where $\varphi _i^{*} Z_i = Z_i \circ \varphi _i \in \Gamma (\varphi _i^{-1} N_i)$ and $\varphi _{i *}$ is section of $Hom(TM_i, \varphi _i^{-1}(TN_i)) = TM_i^{*} \otimes \varphi _i^{-1} (TN_i) \rightarrow M_i = TM_i^{*} \otimes \varphi _i^{-1}(TN_i)$. In addition, the bundle $TM_i^{*} \otimes \varphi _i^{-1}(TN_i)$ has connection $\nabla ^{i}$ induced from the Levi-Civita connection $\nabla ^{M_i}$ of $M_i$ and the pull back connection $\nabla ^{\varphi _i}$. The covariant derivative of $\varphi _{i *}$ called the second fundamental form of $\varphi _i$, i.e. $\nabla ^{i} \varphi _{i *} \in \Gamma (T^{*} M_i \otimes T^{*} M_i \otimes \varphi ^{-1}(TN_i))$ such that

$$\begin{aligned} (\nabla ^{i} \varphi _{i *})(X_i, Y_i) = \nabla _{X_i}^{\varphi _i}(\varphi _{i *}(Y_i)) - \varphi _{i *} (\nabla _{X_i}^{M_i}Y_i), \end{aligned}$$

(26)

where $X_i, Y_i \in \Gamma (TM_i)$. Here the map $\varphi _{*} = (\varphi _1 \times \varphi _2)_{*}$ is section of $Hom(T(M_1 \times M_2), \varphi ^{-1}(T(N_1 \times N_2))) \rightarrow M_1 \times M_2$. Let the bundle $T^{*}(M_1 \times M_2) \otimes \varphi ^{-1}(T(N_1 \times N_2))$ has connection $\nabla$ induced from the Levi-Civita connection $\nabla ^M$ of $M = M_1 \times _f M_2$ and the pull back connection $\nabla ^{\varphi }$. The covariant derivative $\nabla \varphi _*$ is the second fundamental form of $\varphi$, i.e.

$$\begin{aligned}\nabla \varphi _*\in \Gamma (T^{*} (M_1 \times M_2) \otimes T^{*} (M_1 \times M_2) \otimes \varphi ^{-1}(T(N_1 \times N_2))).\end{aligned}$$

Now for $X= (X_1, X_2)$ and $Y= (Y_1, Y_2)$, we have

$$\begin{aligned} (\nabla \varphi _*)(X, Y)=&\nabla _{X}^{\varphi } \varphi _*Y - \varphi _*(\nabla _{X}^M Y) \\=&\nabla _{(X_1, X_2)}^{\varphi } \varphi _*(Y_1, Y_2) - \varphi _*(\nabla _{(X_1, X_2)}^M (Y_1, Y_2)). \end{aligned}$$

Now by using Proposition 4, we get

$$\begin{aligned} (\nabla \varphi _*)(X, Y) =&\nabla _{(X_1+ X_2)}^{\varphi } (\varphi _{1 *} Y_1 + \varphi _ {2 *}Y_2) - \varphi _*(\nabla _{(X_1+ X_2)}^M (Y_1+ Y_2))\nonumber \\=&\nabla _{X_1}^{\varphi } \varphi _{1 *} (Y_1)+ \nabla _{X_2}^{\varphi } \varphi _{1 *} (Y_1)+ \nabla _{X_1}^{\varphi } \varphi _{2 *} (Y_2) \nonumber \\ {}&+ \nabla _{X_2}^{\varphi } \varphi _{2 *} (Y_2)- \varphi _{*}(\nabla _{X_1}^{M} Y_1+ \nabla _{X_1}^{M} Y_2+ \nabla _{X_2}^{M} Y_1+ \nabla _{X_2}^{M} Y_2). \end{aligned}$$

(27)

Using Lemma 2 in (27), we get

$$\begin{aligned} (\nabla \varphi _*)(X, Y)=&\nabla _{X_1}^{\varphi } \varphi _{1 *} (Y_1)+ \nabla _{X_2}^{\varphi } \varphi _{1 *} (Y_1)+ \nabla _{X_1}^{\varphi } \varphi _{2 *} (Y_2) + \nabla _{X_2}^{\varphi } \varphi _{2 *} (Y_2) \\ {}&- \varphi _{*}(\nabla _{X_1}^{M} Y_1+ (X_1(f)/f) Y_2 \\ {}&+ (Y_1(f)/f)X_2 + nor(\nabla _{X_2}^{M} Y_2)+ tan(\nabla _{X_2}^{M} Y_2)) \\=&\nabla _{\varphi _{1 *}X_1}^{N} \varphi _{1 *} Y_1 + \nabla _{\varphi _{2 *} X_2}^{N} \varphi _{1 *} Y_1 + \nabla _{\varphi _{1 *} X_1}^{N} \varphi _{2 *} Y_2 + \nabla _{\varphi _{2 *} X_2}^{N} \varphi _{2 *} Y_2\\ {}&- \varphi _{1*}(\nabla _{X_1}^{M_1} Y_1)- (X_1(f)/f) \varphi _{2 *} Y_2 - (Y_1(f)/f) \varphi _{2 *}X_2 \\ {}&+ g_M(X_2, Y_2) \varphi _{*}(\nabla \ln f) - \varphi _{2 *}(\nabla _{X_2}^{M_2} Y_2). \end{aligned}$$

Using Lemma 9 in above equation, we get

$$\begin{aligned} (\nabla \varphi _*)(X,Y) =&\nabla _{\varphi _{1 *}X_1}^{N_1} \varphi _{1 *} Y_1 + (\varphi _{1 *}Y_1(\rho )/\rho )\varphi _{2 *}X_2 + (\varphi _{1 *}X_1(\rho )/\rho )\varphi _{2 *}Y_2 \nonumber \\ {}&- g_N(\varphi _{2 *} X_2, \varphi _{2 *} Y_2)(\nabla ^N \ln \rho ) + \nabla _{\varphi _{2 *} X_2}^{N_2} \varphi _{2 *} Y_2 - \varphi _{1*}(\nabla _{X_1}^{M_1} Y_1) \nonumber \\ {}&- (X_1(f)/f) \varphi _{2 *} Y_2 - (Y_1(f)/f) \varphi _{2 *}X_2 \\ {}&+ g_M(X_2, Y_2) \varphi _{*}(\nabla ^M \ln f) - \varphi _{2 *}(\nabla _{X_2}^{M_2} Y_2).\nonumber \end{aligned}$$

(28)

This is the second fundamental form (Gauss formula) for a smooth map between Riemannian warped product manifolds. Since $\varphi$ is Riemannian warped product map, (28) implies

$$\begin{aligned} (\nabla \varphi _*)(X,Y) =&\nabla _{\varphi _{1 *}X_1}^{N_1} \varphi _{1 *} Y_1 + (\varphi _{1 *}Y_1(\rho )/\rho )\varphi _{2 *}X_2 + (\varphi _{1 *}X_1(\rho )/\rho )\varphi _{2 *}Y_2 \nonumber \\ {}&- g_M(X_2, Y_2)(\nabla ^N \ln \rho ) + \nabla _{\varphi _{2 *} X_2}^{N_2} \varphi _{2 *} Y_2 - \varphi _{1*}(\nabla _{X_1}^{M_1} Y_1) \nonumber \\ {}&- (X_1(f)/f) \varphi _{2 *} Y_2 - (Y_1(f)/f) \varphi _{2 *}X_2 \\ {}&+ g_M(X_2, Y_2) \varphi _{*}(\nabla ^M \ln f) - \varphi _{2 *}(\nabla _{X_2}^{M_2} Y_2).\nonumber \end{aligned}$$

(29)

We know that f is lift of $\rho$ then by Lemma 1, we have

$$\begin{aligned} \varphi _{*}(\nabla ^{M} \ln f) = \nabla ^{N} \ln \rho . \end{aligned}$$

(30)

Using (26) and (30) in (29), we get the required proof. $\square$

Now, from Proposition 10, Theorem 12 and Lemma 4 of [14] we have following consequences:

Corollary 13

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then

(i):

$(\nabla \varphi _{*})(X, Y) = - \varphi _{1 *}({\mathcal {H}}_1 \nabla _{U_1}^{M_1} V_1) = - \varphi _{1 *}(T_1 (U_1, V_1)) \in \Gamma (range\varphi _{1 *})$ for $X = (U_1, 0),~ Y = (V_1, 0) \in \Gamma (T(M_1 \times M_2))$, where $U_1, V_1 \in \Gamma (ker\varphi _{1 *})$.

(ii):

$(\nabla \varphi _{*})(X, Y) = \nabla _{\varphi _{1 *} X_1}^{N_1} \varphi _{1 *} Y_1 - \varphi _{1 *}(\nabla _{X_1}^{M_1} Y_1) = (\nabla ^1 \varphi _{1 *})(X_1, Y_1) \in \Gamma (range\varphi _{1 *})^\perp$ for $X = (X_1, 0),~ Y = (Y_1, 0) \in \Gamma (T(M_1 \times M_2))$, where $X_1, Y_1 \in \Gamma (ker \varphi _{1 *})^\perp$.

(iii):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^1 \varphi _{1 *})(X_1, Y_1) - \varphi _{2 *}(T_2(U_2, V_2))$ for $X = (X_1, U_2),~ Y = (Y_1, V_2) \in \Gamma (T(M_1 \times M_2))$, where $X_1, Y_1 \in \Gamma (ker \varphi _{1 *})^\perp$ and $U_2, V_2 \in \Gamma (ker \varphi _{2 *})$.

(iv):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^2 \varphi _{2 *})(X_2, Y_2) - \varphi _{1 *}(T_1(U_1, V_1)) - U_1(\ln f) \varphi _{2 *} Y_2 - V_1(\ln f) \varphi _{2 *} X_2$ for $X = (U_1, X_2),~ Y = (V_1, Y_2) \in \Gamma (T(M_1 \times M_2))$, where $U_1, V_1 \in \Gamma (ker \varphi _{1 *})$ and $X_2, Y_2 \in \Gamma (ker \varphi _{2 *})^\perp$.

(v):

$(\nabla \varphi _{*})(X, Y) = - \varphi _{2 *}({\mathcal {H}}_2 \nabla _{U_2}^{M_2} V_2) = - \varphi _{2 *}(T_2 (U_2, V_2)) \in \Gamma (range\varphi _{2 *})$ for $X = (0, U_2), Y = (0, V_2) \in \Gamma (T(M_1 \times M_2))$, where $U_2, V_2 \in \Gamma (ker\varphi _{2 *})$.

(vi):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^2 \varphi _{2 *})(X_2, Y_2) \in \Gamma (range\varphi _{2 *})^\perp$ for $X = (0, X_2),~ Y = (0, Y_2) \in \Gamma (T(M_1 \times M_2))$, where $X_2, Y_2 \in \Gamma (ker \varphi _{2 *})^\perp$.

(vii):

$(\nabla \varphi _{*})(X, Y) = - \varphi _{1 *}(T_1 (U_1, V_1)) - \varphi _{2 *}(T_2 (U_2, V_2)) \in \Gamma (range\varphi _{*})$ for $X = (U_1, U_2), Y = (V_1, V_2) \in \Gamma (T(M_1 \times M_2))$, where $U_i, V_i \in \Gamma (ker\varphi _{i *})$.

(viii):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^1 \varphi _{1 *})(X_1, Y_1) + (\nabla ^2 \varphi _{2 *})(X_2, Y_2) \in \Gamma (range\varphi _{*})^\perp$ for $X = (X_1, X_2),~ Y = (Y_1, Y_2) \in \Gamma (T(M_1 \times M_2))$, where $X_i, Y_i \in \Gamma (ker \varphi _{i *})^\perp$.

(ix):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^1 \varphi _{1 *})(X_1, Y_1) - \varphi _{2 *}({\mathcal {H}}_2 \nabla _{U_2}^{M_2} Y_2)$ for $X = (X_1, U_2),~ Y = (Y_1, Y_2) \in \Gamma (T(M_1 \times M_2))$, where $X_1, Y_1 \in \Gamma (ker \varphi _{1 *})^\perp$, $Y_2 \in \Gamma (ker \varphi _{2 *})^\perp$ and $U_2 \in \Gamma (ker \varphi _{2 *})$.

(x):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^1 \varphi _{1 *})(X_1, Y_1) - \varphi _{2 *}(A_2(X_2, V_2))$ for $X = (X_1, X_2), Y = (Y_1, V_2) \in \Gamma (T(M_1 \times M_2))$, where $X_1, Y_1 \in \Gamma (ker \varphi _{1 *})^\perp$, $X_2 \in \Gamma (ker \varphi _{2 *})^\perp$ and $V_2 \in \Gamma (ker \varphi _{2 *})$.

(xi):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^2 \varphi _{2 *})(X_2, Y_2) - \varphi _{1 *}({\mathcal {H}}_1 \nabla _{U_1} Y_1) - U_1(\ln f) \varphi _{2 *} Y_2$ for $X = (U_1, X_2),~ Y = (Y_1, Y_2) \in \Gamma (T(M_1 \times M_2))$, where $Y_1 \in \Gamma (ker \varphi _{1 *})^\perp$, $X_2, Y_2 \in \Gamma (ker \varphi _{2 *})^\perp$ and $U_1 \in \Gamma (ker \varphi _{1 *})$.

(xii):

$(\nabla \varphi _{*})(X, Y) = (\nabla ^2 \varphi _{2 *})(X_2, Y_2) - V_1(\ln f) \varphi _{2 *} X_2 - \varphi _{1 *} (A_1(X_1, V_1))$ for $X = (X_1, X_2),~ Y = (V_1, Y_2) \in \Gamma (T(M_1 \times M_2))$, where $X_1 \in \Gamma (ker \varphi _{1 *})^\perp$, $X_2, Y_2 \in \Gamma (ker \varphi _{2 *})^\perp$ and $V_1 \in \Gamma (ker \varphi _{1 *})$.

Remark 3

In Lemma 3.1 of [26], Şahin showed that for a Riemannian map $\varphi$ between Riemannian manifolds $(\nabla \varphi _*) (X, Y) \in \Gamma (range \varphi _*)^\perp$ if $X, Y \in \Gamma (ker \varphi _*)^\perp$. Similarly in the (viii) statement of Corollary 13, we get that for a Riemannian warped product map between Riemannian warped product manifolds $(\nabla \varphi _{*})(X, Y) \in \Gamma (range\varphi _{*})^\perp$ for $X = (X_1, X_2),~ Y = (Y_1, Y_2)$, where $X_i, Y_i \in \Gamma (ker \varphi _{i *})^\perp$.

Now, we give the definition of totally geodesic map between Riemannian warped product manifolds.

Definition 3

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then $\varphi$ is called totally geodesic if $(\nabla \varphi _*)(X,Y)= 0$ for all $X= (X_1, X_2),~ Y = (Y_1, Y_2) \in \Gamma (T(M_1 \times M_2))$.

Theorem 14

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then $\varphi$ is totally geodesic if and only if

(i):

$\varphi _1$ is totally geodesic, and

(ii):

$\varphi _2$ is totally geodesic, and

(iii):

f is constant on $ker \varphi _{1 *}$.

Proof

The proof follows by Theorem 12 and Corollary 13. $\square$

5 Harmonic Riemannian Warped Product Maps

In this section, we calculate the tension field for a Riemannian warped product map between Riemannian warped product manifolds and discuss harmonicity [2, 29].

First, we give the definition of the tension field for a smooth map between Riemannian warped product manifolds [2].

Definition 4

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a smooth map between Riemannian warped product manifolds. Then the tension field $\tau (\varphi )$ of $\varphi$ is trace of the second fundamental form of $\varphi$ with respect to $g_M$, i.e.

$$\begin{aligned} \tau (\varphi )= trace (\nabla \varphi _{*})= trace (\nabla \varphi _{*})((X_1,X_2), (Y_1, Y_2))= \sum _{i=1}^{m_1} \sum _{a = m_1 + 1}^{m_1 + m_2} (\nabla \varphi _{*})((e_i, e_a), (e_i, e_a)), \end{aligned}$$

where $(X_1, X_2), (Y_1, Y_2) \in \Gamma (T(M_1 \times M_2))$ and $\{e_i\}_{i = 1}^{m_1}$, $\{e_a\}_{a = m_1 + 1}^{m_1 + m_2}$ are orthonormal bases of $T_{p_1}M_1$ and $T_{p_2}M_2$, respectively. The tension field of $\varphi$ is a vector field along $\varphi$, i.e. $\tau (\varphi ) \in \Gamma _{\varphi } (T(N_1 \times N_2))$.

Lemma 15

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then

$$\begin{aligned} \tau (\varphi ) = -r_1 ~\varphi _{1 *} (H_1) - r_2 ~\varphi _{2 *} (H_2) + (m_1 - r_1) H_3 + (m_2 - r_2) H_4, \end{aligned}$$

where $r_1 = \dim (ker\varphi _{1 *}), r_2 = \dim (ker \varphi _{2 *}), m_1-r_1 = \dim (range\varphi _{1 *})$ and $m_2-r_2 = \dim (range \varphi _{2 *})$. In addition $H_1, H_2, H_3$ and $H_4$ are the mean curvature vector fields of $ker\varphi _{1 *}, ker\varphi _{2 *}$, $range\varphi _{1 *}$ and $range\varphi _{2 *}$, respectively.

Proof

Let $\{u_1,u_2,\dots ,u_{r_1}\}, \{x_{r_1 +1},x_{r_1 + 2},\dots ,x_{m_1}\}, \{u_{r_1 + 1}^*, u_{r_1 + 2}^*, \dots , u_{r_1 + r_2}^*\}$ and $\{x_{m_1 - r_1 +1}^*, x_{m_1 -r_1 +2}^*, \dots , x_{m_1 - r_1 + m_2 - r_2}^*\}$ be orthonormal bases of $ker\varphi _{1 *}, (ker\varphi _{1 *})^\perp$, $ker\varphi _{2 *}$ and $(ker\varphi _{2 *})^\perp$, respectively. We know that

$$\begin{aligned} \tau (\varphi ) = \tau ^{ker\varphi _{*}}(\varphi ) + \tau ^{(ker\varphi _{*})^\perp }(\varphi ). \end{aligned}$$

(31)

Now

$$\begin{aligned} \tau ^{ker\varphi _{*}}(\varphi ) = \sum _{i=1}^{r_1} \sum _{a=r_1 +1}^{r_1 +r_2} (\nabla \varphi _{*}) ((u_i, u_a^*), (u_i, u_a^*)), \end{aligned}$$

(32)

where $(u_i, u_a^*) \in ker\varphi _{1 *} \times ker\varphi _{2 *} = ker\varphi _{*}$, and $\{u_i\}_{i=1}^{r_1}, \{u_a^*\}_{a=r_1 + 1}^{r_1 +r_2}$ are orthonormal bases of $ker\varphi _{1 *}$ and $ker\varphi _{2 *}$, respectively. Using (25) in (32), we get

$$\begin{aligned} \tau ^{ker\varphi _{*}}(\varphi )=&\sum _{i=1}^{r_1} (\nabla ^1 \varphi _{1 *})(u_i, u_i) + \sum _{a = r_1 +1}^{r_1 +r_2} (\nabla ^2 \varphi _{2 *})(u_a^*, u_a^*)\\ {}&+ \sum _{i = 1}^{r_1} \sum _{a = r_1 + 1}^{r_1 + r_2} (\varphi _{1 *}u_i(\ln \rho )) \varphi _{2 *}u_a^* + \sum _{i = 1}^{r_1} \sum _{a = r_1 + 1}^{r_1 + r_2} (\varphi _{1 *} u_i(\ln \rho )) \varphi _{2 *}u_a^* \nonumber \\ {}&- \sum _{i = 1}^{r_1} \sum _{a = r_1 + 1}^{r_1 + r_2} \left\{ (u_i(\ln f)) \varphi _{2 *} u_a^* + (u_i(\ln f)) \varphi _{2 *}u_a^*\right\} . \end{aligned}$$

Using (26) in above equation, we get

$$\begin{aligned} \tau ^{ker\varphi _{*}}(\varphi ) = - \sum _{i = 1}^{r_1} (\nabla ^1 \varphi _{1 *}) (u_i, u_i) - \sum _{a = r_1 + 1}^{r_1 + r_2}(\nabla ^2\varphi _{2 *}) (u_a^*, u_a^*). \end{aligned}$$

(33)

Using (7) and (8) in (33), we get

$$\begin{aligned} \tau ^{ker\varphi _{*}} = - r_1 ~\varphi _{1 *} (H_1) - r_2 ~\varphi _{2 *} (H_2), \end{aligned}$$

(34)

where $H_1$ and $H_2$ are the mean curvature vector fields of $ker\varphi _{1 *}$ and $ker\varphi _{2 *}$ respectively. On the other hand

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi ) = \sum _{i= r_1 + 1}^{m_1} \sum _{a= m_1 - r_1 + 1}^{m_1 -r_1 + m_2 - r_2} (\nabla \varphi _{*})((x_i, x_a^*) (x_i, x_a^*)), \end{aligned}$$

(35)

where $\{x_i\}_{i = r_1 + 1}^{m_1}$ and $\{x_a^*\}_{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2}$ are orthonormal bases of $(ker\varphi _{1 *})^\perp$ and $(ker\varphi _{2 *})^\perp$, respectively. Using (viii) statement of Corollary 13 in (35), we get

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi ) = \sum _{i = r_1 + 1}^{m_1}(\nabla ^1 \varphi _{1 *}) (x_i, x_i) + \sum _{a = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2}(\nabla ^2 \varphi _{2 *}) (x_a^*, x_a^*). \end{aligned}$$

Equivalently

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi )=&\sum _{i= r_1 + 1}^{m_1} \sum _{p=1}^{n_1} g_N ((\nabla ^1 \varphi _{1 *})(x_i, x_i), z_p)z_p\\ {}&+ \sum _{a= m_1 - r_1 + 1}^{m_1 -r_1 + m_2- r_2} \sum _{q=n_1 + 1}^{n_2} g_N ((\nabla ^2 \varphi _{2 *})(x_a^*, x_a^*), z'_q)z'_q, \end{aligned}$$

where $\{z_p\}_{p=1}^{n_1}$ and $\{z'_q\}_{q= n_1 + 1}^{n_1 + n_2}$ are orthonormal bases of $T N_1$ and $T N_2$ respectively. On decomposition

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi )=&\sum _{i = r_1 +1}^{m_1}\sum _{k= r_1 +1}^{m_1} g_{N_1}((\nabla ^1 \varphi _{1 *})(x_i, x_i), {\tilde{e}}_k) {\tilde{e}}_k \\ {}&+ \sum _{i = r_1 +1}^{m_1}\sum _{l=1}^{n_3} g_{N_1}((\nabla ^1 \varphi _{1 *})(x_i, x_i), {\bar{e}}_l), {\bar{e}}_l)\\ {}&+ \sum _{a= m_1 - r_1 +1}^{m_1 - r_1 + m_2 - r_2} \left\{ \rho ^2 \sum _{t= m_1 - r_1 +1}^{m_1 - r_1 + m_2 - r_2} g_{N_2}((\nabla ^2 \varphi _{2 *})(x_a^*, x_a^*), {\hat{e}}_t){\hat{e}}_t \right\} \\ {}&+ \sum _{a= m_1 - r_1 +1}^{m_1 - r_1 + m_2 - r_2} \left\{ \sum _{s=1 + n_3}^{n_3 + n_4}\rho ^2 g_{N_2}((\nabla ^2 \varphi _{2 *})(x_a^*, x_a^*), {\check{e}}_s){\check{e}}_s \right\} , \end{aligned}$$

where $\{{\tilde{e}}_k\}_{k= r_1 + 1}^{m_1}, \{{\bar{e}}_l\}_{l=1}^{n_3}, \{{\hat{e}}_t\}_{t= m_1 - r_1 +1}^{m_1 - r_1 + m_2 - r_2}$ and $\{{\check{e}}_s\}_{s= n_3 + 1}^{n_3 + n_4}$ are orthonormal bases of $range\varphi _{1 *}, (range\varphi _{1 *})^\perp , range\varphi _{2 *}$ and $(range\varphi _{2 *})^\perp$, respectively. Using (26) in above equation, we get

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi ) =&\sum _{i=r_1 +1}^{m_1} \sum _{l=1}^{n_3} g_{N_1} (\nabla _{x_i}^{\varphi _1} \varphi _{1 *}(x_i), {\bar{e}}_l){\bar{e}}_l \nonumber \\ {}&+ \sum _{a=m_1 - r_1 +1}^{m_1 - r_1 + m_2 - r_2} \sum _{s=1 + n_3}^{n_3 + n_4} \rho ^2 g_{N_2} (\nabla _{x_a^*}^{\varphi _2} \varphi _{2 *}(x_a^*), {\check{e}}_s){\check{e}}_s. \end{aligned}$$

(36)

Using (13) and (14) in (36), we get

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi ) = (m_1 - r_1) \sum _{l=1}^{n_3} g_{N_1} (H_3, {\bar{e}}_l){\bar{e}}_l + (m_2 - r_2) \sum _{s=1 + n_3}^{n_3 + n_4} \rho ^2 g_{N_2} (H_4, {\check{e}}_s){\check{e}}_s, \end{aligned}$$

where $H_3$ and $H_4$ are the mean curvature vector fields of $range\varphi _{1 *}$ and $range\varphi _{2 *}$ respectively. Thus

$$\begin{aligned} \tau ^{(ker\varphi _{*})^\perp }(\varphi ) = (m_1 - r_1)H_3 + (m_2 - r_2)H_4. \end{aligned}$$

(37)

Then we get required proof by using (31), (34) and (37). $\square$

Remark 4

In Lemma 4.2 of [27], Şahin showed that for a Riemannian map $\varphi$ between Riemannian manifolds $\tau (\varphi ) = -r \varphi _{*} (H) + (m-r) H'$, where $r = \dim (ker \varphi _*)$ and $m-r = \dim (range \varphi _*)$. Also, H and $H'$ denote the mean curvature vector fields of $ker \varphi _*$ and $range \varphi _*$, respectively. While in Lemma 15, we get that for a Riemannian warped product map between Riemannian warped product manifolds $\tau (\varphi ) = -r_1 \varphi _{1 *} (H_1) - r_2 \varphi _{2 *} (H_2) + (m_1 - r_1) H_3 + (m_2 - r_2) H_4$, where $r_1 = \dim (ker\varphi _{1 *}), r_2 = \dim (ker \varphi _{2 *}), m_1-r_1 = \dim (range\varphi _{1 *})$ and $m_2-r_2 = \dim (range \varphi _{2 *})$. Here $H_1, H_2, H_3$ and $H_4$ are the mean curvature vector fields of $ker\varphi _{1 *}, ker\varphi _{2 *}$, $range\varphi _{1 *}$ and $range\varphi _{2 *}$, respectively.

Now, we give the definition of harmonic map between Riemannian warped product manifolds.

Definition 5

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then $\varphi$ is harmonic if its tension field $\tau (\varphi )$ vanishes.

Theorem 16

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a non-constant Riemannian warped product map between Riemannian warped product manifolds. Then any four conditions imply fifth:

(i):

$\varphi$ is harmonic.

(ii):

The distribution $ker\varphi _{1 *}$ is minimal.

(iii):

The distribution $ker\varphi _{2 *}$ is minimal.

(iv):

The distribution $range\varphi _{1 *}$ is minimal.

(v):

The distribution $range\varphi _{2 *}$ is minimal.

Proof

We know that a distribution is minimal if and only if its mean curvature vector field vanishes. Then the proof follows by Lemma 15. $\square$

Remark 5

The harmonicity conditions for a Riemannian map between Riemannian manifolds were given in Theorem 6.1 of [30] by Şahin. Similarly, we obtain the harmonicity conditions for a Riemannian warped product map between Riemannian warped product manifolds in Theorem 16.

6 Umbilical Riemannian Warped Product Maps

In this section, we construct Weingarten formula for a Riemannian warped product map between Riemannian warped product manifolds and discuss umbilicity.

Let $\varphi = \varphi _1 \times \varphi _2: M = M_1 \times _f M_2 \rightarrow N = N_1 \times _\rho N_2$ be a Riemannian warped product map between Riemannian warped product manifolds and $\nabla ^N$ be the Levi-Civita connection on $N = N_1 \times _\rho N_2$. From now on wards, for the sake of simplicity we denote both the Levi-Civita connection on $(N, g_N)$ and its pullback along $\varphi$ by $\nabla ^N$. Here we denote $(range\varphi _{*})^\perp = (range\varphi _{1 *})^\perp \times (range\varphi _{2 *})^\perp$ is subbundle of $\varphi ^{-1}(T(N_1 \times N_2))$ with fiber $\varphi _{1 *}(T_{p_1}M_1)^\perp \times \varphi _{2 *}(T_{p_2}M_2)^\perp$ the orthogonal complement of $\varphi _{1 *}(T_{p_1}M_1) \times \varphi _{2 *}(T_{p_2}M_2)$ for $g_N = g_{N_1} + \rho ^2 ~g_{N_2}$ over $p = (p_1, p_2) \in M_1 \times _{f} M_2$. For any vector field $X= (X_1, X_2)$ on $M= M_1 \times _f M_2$ and any section $V = (V_1, V_2)$ of $(range\varphi _{*})^\perp = (range\varphi _{1 *})^\perp \times (range\varphi _{2 *})^\perp$, we define $\nabla _X^{{\varphi } \perp } V = \nabla _{X_1 + X_2}^{{\varphi } \perp } {V_1 + V_2}$, which is orthogonal projection of $\nabla _{X_1 + X_2}^{N} {V_1 + V_2}$ on $(range\varphi _{*})^\perp = (range\varphi _{1 *})^\perp \times (range\varphi _{2 *})^\perp$. Then $\nabla ^{{\varphi }\perp }$ is a linear connection on $(range\varphi _{*})^\perp$ such that $\nabla ^{{\varphi }\perp } g_N = 0$ [21].

Now we construct Weingarten formula for Riemannian warped product map. Define shape operator $S_V$ on $range\varphi _{*} = range\varphi _{1 *} \times range\varphi _{2 *}$. Since $\varphi$ is a Riemannian map, for $X = (X_1, X_2) \in \Gamma (ker\varphi _{1 *}) \times \Gamma (ker\varphi _{2 *}) = \Gamma (ker\varphi _{*})^\perp$ and $V = (V_1, V_2) \in \Gamma (range\varphi _{1 *})^\perp \times \Gamma (range\varphi _{2 *})^\perp = \Gamma (range\varphi _{*})^\perp$, we have [29]

$$\begin{aligned} \nabla _{\varphi _{*}{(X_1 + X_2)}}^N (V_1, V_2) = -S_{(V_1, V_2)} \varphi _{*}{(X_1, X_2)} + \nabla _{(X_1, X_2)}^{\varphi \perp } (V_1, V_2). \end{aligned}$$

Since $T_{p_1}M_1 \times T_{p_2}M_2 \cong T_{p_1}M_1 \oplus T_{p_2}M_2$, above equation can be written as

$$\begin{aligned} \nabla _{\varphi _{*}X_1 + \varphi _{*}X_2}^N (V_1 + V_2) = -S_{(V_1 + V_2)} ({\varphi _{*}X_1 + \varphi _{*}X_2}) + \nabla _{(X_1 + X_2)}^{\varphi \perp } (V_1 + V_2). \end{aligned}$$

Since $\nabla$ is a linear connection and $S_{V_1}$ is a shape operator on $range\varphi _{1 *}$, there is no meaning of $S_{V_1} \varphi _{2 *}X_2$. Similarly we treat for $S_{V_2}$. Then by above equation, we have

$$\begin{aligned}&\nabla _{\varphi _{1 *}X_1}^N V_1 + \nabla _{\varphi _{1 *}X_1}^N V_2 + \nabla _{\varphi _{2 *}X_2}^N V_1\\ {}&+ \nabla _{\varphi _{2 *}X_2}^N V_2 = - S_{V_1}\varphi _{1 *}X_1 - S_{V_2}\varphi _{2 *}X_2 \\ {}&+ \nabla _{X_1}^{\varphi \perp } V_1 + \nabla _{X_1}^{\varphi \perp } V_2 + \nabla _{X_2}^{\varphi \perp } V_1 + \nabla _{X_2}^{\varphi \perp } V_2. \end{aligned}$$

Using Lemma 9 in above equation, we get

$$\begin{aligned} \nabla _{\varphi _{1 *}X_1}^{N_1} V_1 + \frac{(\varphi _{1 *}X_1 (\rho ))}{\rho } V_2 + \frac{(V_1(\rho ))}{\rho } \varphi _{2 *}X_2 + \nabla _{\varphi _{2 *}X_2}^{N_2} V_2=&-S_{V_1}\varphi _{1 *}X_1 - S_{V_2}\varphi _{2 *}X_2 \nonumber \\ {}&+ \nabla _{X_1}^{\varphi \perp } V_1 + \nabla _{X_2}^{\varphi \perp } V_2, \end{aligned}$$

(38)

where $S_{V_i}\varphi _{i *}X_i$ is tangential component (vector fields along $\varphi _i$) of $\nabla _{\varphi _{i *}X_i}^{N_i} V_i$. Observe that $\nabla _{\varphi _{i *}X_i}^{N_i} V_i$ is pullback connection of $\nabla ^{N_i}$. Here (38) is known as Weingarten formula for Riemannian warped product map.

Since $(range{\varphi _{*}})^\perp$ is subbundle of $\varphi ^{-1}(T(N \times N_2))$ and $\varphi ^{-1}(T(N_1 \times N_2))$ is bundle on $M = M_1 \times _f M_2$, $(range{\varphi _{*}})^\perp$ is also bundle on M. Thus

$$\begin{aligned}\varphi ^{-1}(T(N_1 \times N_2)) = (range{\varphi _{*}})^\perp \oplus (range\varphi _{*}).\end{aligned}$$

Lemma 17

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a Riemannian warped product map between Riemannian warped product manifolds. Then $\rho$ is a constant function on $(range{\varphi _{*}})^\perp$.

Proof

For $V =(V_1, V_2) \in \Gamma (range{\varphi _{*}})^\perp$, we have

$$\begin{aligned} V(\ln \rho )=&g_{N}(V, \nabla ^N \ln \rho )\\=&g_{N_1}(V_1, \nabla ^{N} \ln \rho ) + \rho ^2 ~g_{N_2}(V_2, \nabla ^{N} \ln \rho ). \end{aligned}$$

Using Lemma 1, we get

$$\begin{aligned} V(\ln \rho ) = g_{N_1}(V_1, \varphi _{1 *}(\nabla ^{M_1}f)). \end{aligned}$$

Since $\varphi _{1 *}(\nabla ^{M_1}f) \in \Gamma (range\varphi _{*})$, $V(\ln \rho ) = 0$. This implies the proof. $\square$

Proposition 18

Let $\varphi = \varphi _1 \times \varphi _2: (M= M_1 \times _f M_2, g_M) \rightarrow (N= N_1 \times _{\rho } N_2, g_N)$ be a non-constant Riemannian warped product map between Riemannian warped product manifolds. Then

$$\begin{aligned} g_{N}(S_{(V_1, V_2)} \varphi _{*}(X_1, X_2), \varphi _{*}(Y_1, Y_2)) = g_{N}((\nabla \varphi _{*})((X_1, X_2), (Y_1, Y_2)), (V_1, V_2)), \end{aligned}$$

where $X = (X_1, X_2)$, $Y = (Y_1, Y_2) \in \Gamma (ker\varphi _{1 *})^\perp \times \Gamma (ker\varphi _{2 *})^\perp$ and $V = (V_1, V_2) \in \Gamma (range\varphi _{1 *})^\perp \times \Gamma (range\varphi _{2 *})^\perp$.

Proof

By Weingarten formula for $X = (X_1, X_2)$, $Y = (Y_1, Y_2) \in \Gamma (ker\varphi _{1 *})^\perp \times \Gamma (ker\varphi _{2 *})^\perp$ and $V = (V_1, V_2) \in \Gamma (range\varphi _{*})^\perp$, we have

$$\begin{aligned}&g_N(S_{V_1}{\varphi _{1 *}X_1} + S_{V_2}{\varphi _{2 *}X_2}, ~ \varphi _{1 *}Y_1 + \varphi _{2 *}Y_2) \nonumber \\ {}&\quad = g_N( \nabla _{X_1}^{\varphi \perp } V_1 + \nabla _{X_2}^{\varphi \perp } V_2 -\nabla _{\varphi _{1 *}X_1}^N V_1 - \nabla _{\varphi _{2 *}X_2}^N V_2 \\ {}&\qquad - (\varphi _{1 *}X_1 (\ln \rho ))V_2 - (V_1(\ln \rho )) \varphi _{2 *}X_2, ~ {\varphi _{1 *}Y_1 + \varphi _{2 *}Y_2}),\nonumber \end{aligned}$$

(39)

which implies

$$\begin{aligned} g_N(S_{V_1}{\varphi _{1 *}X_1} + S_{V_2}{\varphi _{2 *}X_2}, ~ \varphi _{1 *}Y_1 + \varphi _{2 *}Y_2)=&-g_N(\nabla _{\varphi _{1 *}X_1}^{N_1} V_1, ~\varphi _{1 *}Y_1) \\ {}&- g_N(\nabla _{\varphi _{2 *}X_2}^{N_2} V_2, ~\varphi _{2 *}Y_2)\\ {}&- g_N((\varphi _{1 *}X_1)(\ln \rho )V_2, ~\varphi _{1 *}Y_1 + \varphi _{2 *}Y_2)\\ {}&- g_N(V_1(\ln \rho )\varphi _{2 *}X_2, ~\varphi _{1 *}Y_1 + \varphi _{2 *}Y_2) \\=&- g_{N_1}(\nabla _{\varphi _{1 *}X_1}^{N_1} V_1, ~\varphi _{1 *}Y_1)\\ {}&- \rho ^2 g_{N_2}(\nabla _{\varphi _{2 *}X_2}^{N_2} V_2, ~\varphi _{2 *}Y_2)\\ {}&- \rho ^2 \left\{ V_1(\ln \rho ) g_{N_2}(\varphi _{2 *}X_2, \varphi _{2 *}Y_2) \right\} . \end{aligned}$$

Using metric compatibility condition and Lemma 17 in above equation, we get

$$\begin{aligned} g_N(S_{V_1}{\varphi _{1 *}X_1} + S_{V_2}{\varphi _{2 *}X_2}, ~ \varphi _{1 *}Y_1 + \varphi _{2 *}Y_2)=&g_{N_1}(\nabla _{\varphi _{1 *}X_1}^{N_1} \varphi _{1 *}Y_1, V_1) \nonumber \\ {}&+ \rho ^2 ~g_{N_2}(\nabla _{\varphi _{2 *}X_2}^{N_2} \varphi _{2 *}Y_2, V_2). \end{aligned}$$

(40)

Using (4) in (40), we get

$$\begin{aligned} g_N(S_{V_1}{\varphi _{1 *}X_1} + S_{V_2}{\varphi _{2 *}X_2}, ~ \varphi _{1 *}Y_1 + \varphi _{2 *}Y_2) =&g_{N_1}(\nabla _{X_1}^{\varphi _1} \varphi _{1 *}(Y_1), V_1) \\ {}&+ \rho ^2 g_{N_2}(\nabla _{X_2}^{\varphi _2} \varphi _{2 *}(Y_2), V_2). \end{aligned}$$

Using (26) in above equation, we get

$$\begin{aligned} g_N(S_{V_1}{\varphi _{1 *}X_1} + S_{V_2}{\varphi _{2 *}X_2}, ~ \varphi _{1 *}Y_1 + \varphi _{2 *}Y_2)=&g_{N_1}((\nabla ^1 \varphi _{1 *})(X_1, Y_1), V_1) \nonumber \\ {}&+ \rho ^2 ~g_{N_2}((\nabla ^2 \varphi _{2 *})(X_2, Y_2), V_2). \end{aligned}$$

(41)

This implies the proof. $\square$

Remark 6

For a Riemannian map between Riemannian manifolds Şahin obtained that $g_{N}(S_{V} \varphi _{*}X, \varphi _{*}Y) = g_{N}((\nabla \varphi _{*})(X, Y), V)$ for $X, Y \in \Gamma (ker\varphi _*)^\perp$ and $V \in \Gamma (range \varphi _*)^\perp$ [29]. While in Proposition 18, for a Riemannian warped product map between Riemannian warped product manifolds we obtain that

$g_{N}(S_{(V_1, V_2)} \varphi _{*}(X_1, X_2), \varphi _{*}(Y_1, Y_2)) = g_{N}((\nabla \varphi _{*})((X_1, X_2), (Y_1, Y_2)), (V_1, V_2))$ for $X = (X_1, X_2)$, $Y = (Y_1, Y_2) \in \Gamma (ker\varphi _{1 *})^\perp \times \Gamma (ker\varphi _{2 *})^\perp$ and $V = (V_1, V_2) \in \Gamma (range\varphi _{1 *})^\perp \times \Gamma (range\varphi _{2 *})^\perp$. Now, since the second fundamental form $(\nabla \varphi _{*})((X_1, X_2), (Y_1, Y_2))$ of $\varphi$ is symmetric, we conclude that $S_V$ is a symmetric linear transformation of $(range\varphi _{*})$.

Now we give definition of umbilical map between Riemannian warped product manifolds.

Definition 6

Let $\varphi = \varphi _1 \times \varphi _2: M = M_1 \times _f M_2 \rightarrow N = N_1 \times _\rho N_2$ be a Riemannian warped product map between Riemannian warped product manifolds. Then we say that $\varphi$ is an umbilical Riemannian warped product map at $p = (p_1, p_2) \in M_1 \times _f M_2$ if

$$\begin{aligned} S_{(V_1, V_2)} \varphi _{*(X_1, X_2)} = \lambda (\varphi _{*}(X_1, X_2)). \end{aligned}$$

Equivalently

$$\begin{aligned} S_{(V_1 + V_2)} ({\varphi _{1 *}X_1 + \varphi _{2 *}X_2}) = \lambda (\varphi _{1 *}X_1 + \varphi _{2 *}X_2), \end{aligned}$$

where $\lambda : N = N_1 \times _\rho N_2 \rightarrow {\mathbb {R}}$ is a smooth function such that $\lambda = (\lambda _1, \lambda _2)$ and $\lambda _i$ is smooth function on $N_i$. We say $\varphi$ is an umbilical map if it is umbilical at all $p \in M$.

Theorem 19

Let $\varphi = \varphi _1 \times \varphi _2: M = M_1 \times _f M_2 \rightarrow N = N_1 \times _\rho N_2$ be a Riemannian warped product map between Riemannian warped product manifolds. Then $\varphi$ is umbilical if and only if

$$\begin{aligned} (\nabla \varphi _{*})((X_1, X_2), (Y_1, Y_2)) = (H_3, H_4) g_{M}((X_1, Y_1), (X_2, Y_2)), \end{aligned}$$

where $X = (X_1, X_2), Y = (Y_1, Y_2) \in \Gamma (ker\varphi _{1 *})^\perp \times \Gamma (ker\varphi _{2 *})^\perp$ and $V = (V_1, V_2) \in \Gamma (range\varphi _{1 *})^\perp \times \Gamma (range\varphi _{2 *})^\perp$. In addition, $H_3$ and $H_4$ are the mean curvature vector fields of $range\varphi _{1 *}$ and $range\varphi _{2 *}$ respectively.

Proof

Let $\{u_1,u_2,\dots ,u_{r_1}\}, \{x_{r_1 +1},x_{r_1 + 2},\dots ,x_{m_1}\}, \{u_{r_1 + 1}^*, u_{r_1 + 2}^*, \dots , u_{r_1 + r_2}^*\}$ and $\{x_{m_1 - r_1 +1}^*, x_{m_1 -r_1 +2}^*, \dots , x_{m_1 - r_1 + m_2 - r_2}^*\}$ be orthonormal bases of $ker\varphi _{1 *}, (ker\varphi _{1 *})^\perp$, $ker\varphi _{2 *}$ and $(ker\varphi _{2 *})^\perp$, respectively. Then the Riemannian warped product map $\varphi = \varphi _1 \times \varphi _2$ implies $\{\varphi _{1 *}(x_{r_1 + 1}), ~\varphi _{1 *}(x_{r_1 + 2}),\dots ,\varphi _{1 *}(x_{m_1})\}$ and $\{\varphi _{2 *}(x^*_{m_1 - r_1 + 1}), \dots$, $\varphi _{2 *}(x^*_{m_1 - r_1 + m_2 - r_2})\}$ are orthonormal bases of $range\varphi _{1 *}$ and $range\varphi _{2 *}$, respectively. Then

$$\begin{aligned}{} & {} \sum _{i = r_1 + 1}^{m_1} \sum _{j = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} g_N(S_{(V_1 + V_2)} \varphi _{*}(x_i, x^*_j), \varphi _{*}(x_i, x^*_j)) \\{} & {} \quad = \sum _{i = r_1 + 1}^{m_1} \sum _{j = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} g_N(\lambda \{\varphi _{1 *}(x_i) + \varphi _{2 *}(x^*_j)\}, ~\varphi _{1 *}(x_i) + \varphi _{2 *}(x^*_j)). \end{aligned}$$

On solving above equation, we get

$$\begin{aligned}&\sum _{i = r_1 + 1}^{m_1} \sum _{j = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} g_N(S_{V_1}{(\varphi _{1 *}x_i)} + S_{V_2}{(\varphi _{2 *}x^*_j), ~\varphi _{1 *}(x_i) + \varphi _{2 *}(x^*_j)}) \nonumber \\ {}&\quad = \sum _{i = r_1 + 1}^{m_1} \sum _{j = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} (\lambda _1, \lambda _2)(g_{N_1}(\varphi _{1 *} x_i, \varphi _{1 *} x_i) + \rho ^2 ~g_{N_2}(\varphi _{1 *} x^*_j, \varphi _{1 *} x^*_j)). \end{aligned}$$

(42)

Using (41) in (42), we get

$$\begin{aligned}&\sum _{i = r_1 + 1}^{m_1} g_{N_1}((\nabla ^1\varphi _{1 *})(x_i, x_i), V_1) + \sum _{j = m_1 - r_1 + 1}^{m_1 - r_1 + m_2 - r_2} \rho ^2 ~g_{N_2}((\nabla ^2\varphi _{2 *})(x^*_j, x^*_j), V_2) \\ {}&\quad = \lambda _1 (m_1 - r_1) + \lambda _2 ~\rho ^2 ~(m_2 - r_2). \end{aligned}$$

Using (13) and (14) in above equation, we get

$$\begin{aligned}&(m_1 - r_1) ~g_{N_1} (H_3, V_1) + \rho ^2 ~(m_2 - r_2) ~g_{N_2} (H_4, V_2)\\&= \lambda _1 ~(m_1 - r_1) + ~\lambda _2 ~\rho ^2 ~(m_2 - r_2), \end{aligned}$$

where $H_3$ and $H_4$ are the mean curvature vector fields of $range\varphi _{1 *}$ and $range\varphi _{2 *}$ respectively. On comparison, we get

$$\begin{aligned} \lambda _1 = g_{N_1}(H_3, V_1)~\text {and}~\lambda _2 = g_{N_2}(H_4, V_2), \end{aligned}$$

By (41) and Definition 6, we get

$$\begin{aligned}&g_{N_1}((\nabla ^1\varphi _{1 *})(X_1, Y_1), V_1) + \rho ^2 ~g_{N_2}((\nabla ^2\varphi _{2 *})(X_2, Y_2), V_2)\\ {}&\quad = g_{N_1}(H_3, V_1) g_{M_1}(X_1, Y_1) + ~\rho ^2 ~g_{N_2}(H_4, V_2) ~g_{M_2}(X_2, Y_2), \end{aligned}$$

which implies

$$\begin{aligned}&g_N((\nabla \varphi _{*})((X_1, X_2)(Y_1, Y_2)), (V_1, V_2))\\&= g_N((H_3, H_4), (V_1, V_2)) g_{M}((X_1, Y_1), (X_2, Y_2)). \end{aligned}$$

This implies the required proof. $\square$

Remark 7

In Lemma 4.1 of [28], Şahin showed that a Riemannian map $\varphi$ between Riemannian manifolds is umbilical if and only if $(\nabla \varphi _{*})(X, Y) = H' g_{M}(X, Y)$ for $X, Y \in \Gamma (ker\varphi _*)^\perp$ and $H'$ mean curvature vector field of $range\varphi _*$. While in Theorem 19, we show that a Riemannian warped product map between Riemannian warped product manifolds is umbilical if and only if $(\nabla \varphi _{*})((X_1, X_2), (Y_1, Y_2)) = (H_3, H_4) g_{M}((X_1, Y_1), (X_2, Y_2))$ for $X = (X_1, X_2), Y = (Y_1, Y_2) \in \Gamma (ker\varphi _{1 *})^\perp \times \Gamma (ker\varphi _{2 *})^\perp$, and $H_3$ and $H_4$ the mean curvature vector fields of $range\varphi _{1 *}$ and $range\varphi _{2 *}$ respectively.