On Chen Ideal Submanifolds Satisfying Some Conditions of Pseudo-symmetry Type

Abstract

In this paper, we study Chen ideal submanifolds $M^n$ of dimension n in Euclidean spaces $\mathbb {E}^{n+m}$ ($n \ge 4$, $m \ge 1$) satisfying curvature conditions of pseudo-symmetry type of the form: the difference tensor $R \cdot C - C \cdot R$ is expressed by some Tachibana tensors. Precisely, we consider one of the following three conditions: $R \cdot C - C \cdot R$ is expressed as a linear combination of Q(g , R) and Q(S , R) , $R \cdot C - C \cdot R$ is expressed as a linear combination of Q(g , C) and Q(S , C) and $R \cdot C - C \cdot R$ is expressed as a linear combination of $Q(g , g \wedge S)$ and $Q(S , g \wedge S)$. We then characterize Chen ideal submanifolds $M^n$ of dimension n in Euclidean spaces $\mathbb {E}^{n+m}$ ($n \ge 4$, $m \ge 1$) which satisfy one of the following six conditions of pseudo-symmetry type: $R \cdot C - C \cdot R$ and Q(g , R) are linearly dependent, $R \cdot C - C \cdot R$ and Q(S , R) are linearly dependent, $R \cdot C - C \cdot R$ and Q(g , C) are linearly dependent, $R \cdot C - C \cdot R$ and Q(S , C) are linearly dependent, $R \cdot C - C \cdot R$ and $Q(g , g \wedge S)$ are linearly dependent and $R \cdot C - C \cdot R$ and $Q(S , g \wedge S)$ are linearly dependent. We also prove that the tensors $R \cdot R - Q(S,R)$ and Q(g, C) are linearly dependent at every point of $M^n$ at which its Weyl tensor C is non-zero.

1 Some Generalized Einstein Metric Conditions

As it was presented in [61]: Elie Cartan in his book [5] defined the axiom of r-planes as follows: a Riemannian manifold M of dimension $n > 3$ satisfies the axiom of r-planes, where r is a fixed integer $2 < r < n$, if for each point p of M and any r-dimensional subspace S of the tangent space $T_{p}(M)$ there exists an r-dimensional totally geodesic submanifold V containing p such that $T_{p} (V) = S$. He proved that if M satisfies the axiom of r-planes for some r, then M has constant sectional curvature [5]. In [61] it was proposed the following axiom called axiom of r-spheres: for each point p of M and any r-dimensional subspace S of $T_{p}(M)$, there exists an r-dimensional umbilical submanifold V with parallel mean curvature vector field such that $p \in V$ and $T_{p}(V) = S$. In [61] (Theorem) it was proved that a Riemannian manifold M of dimension $n > 3$ satisfies the axiom of r-spheres for some r, $2 < r < n$, then M has constant sectional curvature. Further, axioms of this kind (i.e. related to properties of submanifolds) were introduced and investigated by several authors, e.g. see [67]. Reference [7, Chap. 3, Sect. 20] contains a survey related to this subject. For recent results we refer to [59] and [66] and references therein.

Other kind of investigations on submanifolds in Riemannian manifolds was proposed by Bang-Yen Chen in the early 1990s, introducing a family of Riemannian invariants $\delta (n_{1}, \ldots , n_{k})$, known also as the $\delta$-invariants, $\delta$-curvatures or Chen invariants. At the same time he established for arbitrary Riemannian submanifolds general optimal inequalities involving those new intrinsic invariants (cf. [16]). As it was stated in [12]: the $\delta$-curvatures are very different in nature from the standard scalar and Ricci curvatures; simply due to the fact that both scalar and Ricci curvatures are the “total sum” of sectional curvatures on a Riemannian manifold. In contrast, the $\delta$-curvature invariants are obtained from the scalar curvature by throwing away a certain amount of sectional curvatures. In this way we can obtain other invariants also called $\delta$-invariants [9, p. 253]: Kählerian $\delta$-invariants (see, e.g. [13]), affine $\delta$-invariants [14], contact $\delta$-invariants [16], submersion $\delta$-invariant [8, 11], etc. We mention that in [15], by an application of some $\delta$-invariants, a characterizations of Einstein spaces and conformally flat spaces were found, generalizing two well-known results of I.M. Singer - J.A. Thorpe and of R.S. Kulkarni. $\delta$-invariants were investigated by several authors. We refer to [6, 8] and [9] as fundamental works on $\delta$-invariants. We also refer to recent survey articles [10] and [11] related to that subject.

Our paper is related to the above-mentioned Chen’s theory. Namely, we investigate curvature properties of pseudo-symmetry type of submanifolds $M^{n}$ in Euclidean ambient spaces $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$, which realizes an optimal equality between their squared mean curvature, i.e. the extrinsic scalar-valued curvature, and their $\delta$-curvature, more precisely, $\delta (2)$-curvature of Chen, which is one of the main intrinsic scalar-valued curvature invariants. Submanifolds having that property are called Chen ideal submanifolds.

Let (M, g), $\dim M = n \ge 3$, be a semi-Riemannian manifold and let $\nabla$ be its Levi-Civita connection. The manifold (M, g) is said to be an Einstein manifold [3] if at every point of M its Ricci tensor S is proportional to the metric tensor g, i.e.

$$\begin{aligned} S= & {} \frac{\tau }{n}\, g \end{aligned}$$

(1)

on M, where $\tau$ is the scalar curvature of (M, g). In particular, if S vanishes on M then (M, g) it is called a Ricci-flat manifold. According to [3, p. 432], the condition (1) is called the Einstein metric condition. Evidently, if a manifold (M, g), $n \ge 3$, is a non-Einstein manifold then the set ${\mathcal {U}}_{S}$ of all points at which S is not proportional to g is an open and non-empty subset of M. Further, (M, g) is said to be a quasi-Einstein manifold if at every point $p \in {\mathcal {U}}_{S}$ we have $\text{ rank }\, (S - \alpha \, g) = 1$, for some $\alpha \in \mathbb {R}$, i.e. $S = \alpha \, g + \varepsilon \, w \otimes w$, for some $\alpha \in \mathbb {R}$, where $\varepsilon = \pm 1$ and w is a non-zero covector at p. It is known that quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations and investigation on quasi-umbilical hypersurfaces of conformally flat spaces (e.g. see [18, 26], and references therein).

An extension of the class of Einstein semi-Riemannian manifolds form manifolds with parallel Ricci tensor S, i.e. $\nabla S = 0$. Such manifolds are called Ricci-symmetric. A very important subclass of the class of Ricci-symmetric manifolds forms locally symmetric manifolds, i.e. manifolds with parallel Riemann–Christoffel curvature tensor R, i.e. $\nabla R = 0$. This implies the following integrability condition

$$\begin{aligned} {\mathcal {R}}(X,Y ) \cdot R= & {} 0 , \end{aligned}$$

(2)

where ${\mathcal {R}}(X,Y ) \cdot$ denotes the derivation obtained from the curvature endomorphism ${\mathcal {R}}(X,Y )$ and X, Y are vector fields on M. From (2) we get immediately

$$\begin{aligned} {\mathcal {R}}(X,Y ) \cdot C= & {} 0 , \end{aligned}$$

(3)

where C is the Weyl conformal curvature tensor of (M, g). We refer to Sect. 2 for precise definitions of the symbols used. Manifolds satisfying (2), resp. (3), are called semi-symmetric manifolds [64], resp. Weyl-semi-symmetric manifolds [23]. We denote by ${\mathcal {U}}_{C}$ the set of all points of a semi-Riemannian manifold (M, g), $n \ge 4$, at which its Weyl conformal curvature tensor C is non-zero. In [52] it was proved that (2) and (3) are equivalent at every point of ${\mathcal {U}}_{C}$ of a manifold (M, g), $n \ge 5$. That result is not true when $n=4$ [20, 71]. We also mention that hypersurfaces satisfying (3) or

$$\begin{aligned} {\mathcal {C}}(X,Y ) \cdot R= & {} 0 , \end{aligned}$$

(4)

were investigated in [4]. ${\mathcal {C}}(X,Y ) \cdot$ denotes the derivation obtained from the Weyl conformal curvature endomorphism ${\mathcal {C}}(X,Y )$.

An extension of the class of semi-symmetric, resp. Weyl-semi-symmetric, manifolds form pseudo-symmetric, resp. Weyl-pseudo-symmetric, manifolds.

A semi-Riemannian manifold (M, g), $n \ge 3$, is said to be pseudo-symmetric [23, 26, 34, 35] if the tensors ${\mathcal {R}}(X,Y ) \cdot R$ and $(X \wedge _{g} Y ) \cdot R$ are linearly dependent at every point of M. This is equivalent to

$$\begin{aligned} {\mathcal {R}}(X,Y ) \cdot R= & {} L_{R}\, (X \wedge _{g} Y ) \cdot R \end{aligned}$$

(5)

on ${\mathcal {U}}_{R} = \{x \in M\, | \, R - (\kappa /((n-1)n))\, G \ne 0\ \text{ at }\ x \}$, where $L_{R}$ is some function on this set and the (0, 4)-tensor G is defined by $G(X,Y,W,Z) = g( (X \wedge _{g} Y) Z, W)$. It is easy to see that the function $L_{R}$ is uniquely determined on ${\mathcal {U}}_{R}$. We note that ${\mathcal {U}}_{S} \cup {\mathcal {U}}_{C} = {\mathcal {U}}_{R}$.

In [46] it was shown that hypersurfaces in spaces of constant curvature, with exactly two distinct principal curvatures at every point, are pseudo-symmetric. Thus in particular, Cartan’s and Schouten’s investigations of quasi-umbilical hypersurfaces in spaces of constant curvature are closely related to pseudo-symmetric manifolds (see [35]). It is clear that every semi-symmetric manifold is pseudo-symmetric. However, the converse statement is not true. For instance, the Schwarzschild spacetime, the Kottler spacetime and the Reissner–Nordström spacetime satisfy (5) with non-zero function $L_{R}$ [45] (see also [53]). The Schwarzschild spacetime was discovered in 1916 by K. Schwarzschild, during his study on solutions of Einstein’s equations. It seems that the Schwarzschild spacetime is the “oldest” example of a non-semi-symmetric, pseudo-symmetric warped product (see [35]). A similar remark is related to Friedmann–Lemaître–Robertson–Walker spacetimes (cf. [35]). We refer to [23, 35, 54] and [55] for a more detailed presentation on the class of pseudo-symmetric manifolds. A geometric interpretation of the notion of the pseudo-symmetry is given in [54], see also [55].

A semi-Riemannian manifold (M, g), $n \ge 4$, is said to be Weyl-pseudo-symmetric [23, 26, 35] if the tensors ${\mathcal {R}}(X,Y ) \cdot C$ and $(X \wedge _{g} Y ) \cdot C$ are linearly dependent at every point of M. This is equivalent to

$$\begin{aligned} {\mathcal {R}}(X,Y ) \cdot C= & {} L\, (X \wedge _{g} Y ) \cdot C \end{aligned}$$

(6)

on ${\mathcal {U}}_{C}$, where L is some function on this set. The function L is uniquely determined on ${\mathcal {U}}_{C}$. A geometric interpretation of the notion of the Weyl-pseudo-symmetry is given in [58]. Every pseudo-symmetric manifold is Weyl-pseudo-symmetric. The converse statement is not true. Precisely, (5) and (6) are equivalent at every point of ${\mathcal {U}}_{C}$ of a manifold (M, g), $n \ge 5$, and that result is not true when $n=4$, see [35] and references therein.

A semi-Riemannian manifold (M, g), $n \ge 4$, is said to be a manifold with pseudo-symmetric Weyl tensor [23, 26, 35, 47] if the tensors ${\mathcal {C}}(X,Y) \cdot C$ and $(X \wedge _{g} Y ) \cdot C$ are linearly dependent at every point of M. This is equivalent to

$$\begin{aligned} {\mathcal {C}}(X,Y) \cdot C= & {} L_{C}\, (X \wedge _{g} Y ) \cdot C \end{aligned}$$

(7)

on${\mathcal {U}}_{C}$, where $L_{C}$ is some function on this set. The function L is uniquely determined on ${\mathcal {U}}_{C}$. It is clear that (7) is invariant under the conformal deformations of the metric tensor g. We say that (5), (6) and (7) are pseudo-symmetry-type curvature conditions [23, 26, 35, 55]. In Sect. 3 we present more information on pseudo-symmetric manifolds, as well as manifolds with pseudo-symmetric Weyl tensor.

In what follows, for a (0, k)-tensor T and a symmetric (0, 2)-tensor A on a manifold (M, g) we will denote the tensors ${\mathcal {R}}(X,Y) \cdot T$, ${\mathcal {C}}(X,Y) \cdot T$ and $(X \wedge _{A} Y) \cdot T$ by $R \cdot T$, $C \cdot T$ and Q(A, T), respectively. The tensor Q(A, T) is called the -Tachibana tensor (e.g. see [33]). In particular, we have the following (0, 6)-tensors: $R\cdot R$, $R\cdot C$, $C \cdot R$, $C \cdot C$ and $R \cdot C - C \cdot R$, and the (0, 6)-Tachibana tensors: Q(g, R), Q(S, R), Q(g, C), Q(S, C), $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$. The tensor $R \cdot C - C \cdot R$ is called the difference tensor. Now we can present (5) and (7) in the form

$$\begin{aligned} R \cdot R= & {} L_{R}\, Q(g,R), \end{aligned}$$

(8)

$$\begin{aligned} C \cdot C= & {} L_{C}\, Q(g,C), \end{aligned}$$

(9)

respectively. We also note that $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$ can be expressed by some other Tachibana tensors (e.g. see [33], p. 228)

$$\begin{aligned} \begin{aligned}&Q(g,g \wedge S) = - Q(S, G) ,\\&Q(S,g \wedge S) = - \frac{1}{2}\, Q(g, S \wedge S) . \end{aligned} \end{aligned}$$

(10)

Let (M, g), $n \ge 4$, be a semi-Riemannian manifold. Trivially, if $R \,\cdot \, C = C \cdot R = 0$ then $R\, \cdot \, C - C \cdot R = 0$. Conversely, if $R\cdot C - C \cdot R$ is a zero tensor on ${\mathcal {U}} = {\mathcal {U}}_{S}\, \cap \, {\mathcal {U}}_{C} \subset M$ then $C \cdot R = 0$ and $R \cdot R = 0$ (and in a consequence $R \cdot C = 0$) on ${\mathcal {U}}$ [41, Corollary 4.1]. It is also clear that the difference tensor $R\cdot C - C \cdot R$ vanishes identically on any Ricci-flat manifold. However, $R\cdot C - C \cdot R$ is a non-zero tensor on every non-Ricci-flat Einstein manifold. This is a consequence of the fact that, on every Einstein manifold (M, g), $n \ge 4$, the following identity is satisfied [41, Theorem 3.1]

$$\begin{aligned} R\cdot C - C \cdot R = \frac{\tau }{(n-1)n}\, Q(g,R) . \end{aligned}$$

(11)

We note that on any Einstein manifold (M, g), $n \ge 4$, we have

$$\begin{aligned} \frac{\tau }{n}\, Q(g,R)= & {} Q(S,R) \ =\ Q(S,C) \ =\ \frac{\tau }{n}\, Q(g,C) ,\nonumber \\ Q(g,g \wedge S)= & {} Q(S, g \wedge S)\ =\ 0 . \end{aligned}$$

(12)

Now from (11) and (12) it follows that on every Einstein manifold (M, g), $n \ge 4$, we have

$$\begin{aligned} R\cdot C - C \cdot R= & {} \frac{\tau }{(n-1)n}\, Q(g,C)\ =\ \frac{1}{n-1}\, Q(S,R)\ =\ \frac{1}{n-1}\, Q(S,C) . \end{aligned}$$

We also can investigate curvature properties of non-Einstein and non-conformally flat semi-Riemannian manifolds of dimension $\ge$4 satisfying the following condition.

($*$) the difference tensor $R\cdot C - C \cdot R$ is a linear combination of the Tachibana tensors: Q(g, R), Q(g, C), Q(S, R), Q(S, C), $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$.

A survey of results on semi-Riemannian manifolds satisfying ($*$) is given in [26]. Furthermore, results of [28] show that some particular cases of ($*$) are realized on hypersurfaces in space forms. For instance

$$\begin{aligned} R\cdot C - C \cdot R= & {} \frac{1}{n-2}\, Q(S,R) - \frac{2}{n-1}\, Q(g,R) \end{aligned}$$

on the Cartan hypersurfaces in $S^{n+1}(1)$, $n = 6, 12, 24$ (see [28], Theorem 1.3, and references therein). For recent results on hypersurfaces in space forms satisfying particular cases of ($*$) we refer to [42]. We also mention that hypersurfaces in space forms for which the tensor $R \cdot C$ or the tensor $C \cdot R$ is a linear combinatin of the tensors Q(S, R), Q(g, R), $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$ and were investigated in [33, 50] and [63].

As we presented above, some particular cases of ($*$) are realized on Einstein manifolds. Therefore ($*$) is called a generalized Einstein metric condition. Clearly, ($*$) is also a condition of pseudo-symmetry type. A presentation of results on Riemannian manifolds satisfying certain generalized Einstein metric conditions is given in [3].

We present now some results on semi-Riemannian manifolds satisfying the following conditions:

• (i) $R \cdot C - C \cdot R$ and Q(g , R) are linearly dependent,

• (ii) $R \cdot C - C \cdot R$ and Q(S , R) are linearly dependent,

• (iii) $R \cdot C - C \cdot R$ and Q(g , C) are linearly dependent,

• (iv) $R \cdot C - C \cdot R$ and Q(S , C) are linearly dependent,

• (v) $R \cdot C - C \cdot R$ and $Q(g , g \wedge S)$ are linearly dependent,

• (vi) $R \cdot C - C \cdot R$ and $Q(S , g \wedge S)$ are linearly dependent.

Manifolds satisfying (i) and (ii) were investigated in [38] and [41], respectively. Examples of warped product manifolds satisfying (i), resp., (ii), are given in [27], resp., in [38] and [60]. Further, it seems that there is no essential result on manifolds satisfying (vi) with non-zero tensors $R\cdot C - C \cdot R$, $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$. Manifolds satisfying (iii) or (vi) were investigated in [29]. Next, manifolds satisfying (v) were investigated in [1] and [57]. In particular, examples of warped product manifolds satisfying that condition are given in [1]. Section 6 of [26] contains some results on manifolds satisfying (iv). An example of a warped product manifold satisfying (iv) is given in [39, Example 5.1]. For further results on manifolds satisfying (iv) we refer to [29]. We also mention that warped product manifolds satisfying (i), (ii), (iv) and (v) are quasi-Einstein or not. Recently some curvature properties of manifolds satisfying these conditions were obtained in [30].

Pseudo-symmetric Chen ideal submanifolds $M^{n}$ of dimension n in Euclidean spaces $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$, were investigated in [32] and [43]. In particular, in Sect. 3 of [32] it was stated that non-Einstein and non-conformally flat pseudo-symmetric Chen ideal submanifolds M are Roter spaces. The difference tensor $R\cdot C - C \cdot R$ of a Roter space is a linear combination of the tensors $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$ (e.g. see [28, Proposition 4.2]). In Sects. 3 and 4 we present more facts related to Chen ideal submanifolds and Roter spaces. In Sect. 3 we also recall that Chen ideal submanifolds $M^{n}$ of codimension m in $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$, satisfy (7) [32, 43]. Moreover, in that section we prove that on the set $U_{C}$ of every Chen ideal submanifold $M^{n}$ of dimension n in $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$, we have

$$\begin{aligned} R \cdot R - Q(S,R)= & {} L\, Q(g,C) , \end{aligned}$$

where L is some function on this set. We mention that the last equation is satisfied on any hypersurface in space forms (e.g. see [28, Eq. (22)]). In particular, the tensor $R \cdot R - Q(S,R)$ vanishes on any hypersurface in a semi-Euclidean space.

With respect to the above presentation, in this paper we investigate Chen ideal submanifolds satisfying some particular cases of ($*$). Precisely, we investigate Chen ideal submanifolds $M^{n}$ in $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$, satisfying:

$$\begin{aligned} R \cdot C - C \cdot R= & {} L_1 Q(g , R) + L_2 Q(S , R),\\ R \cdot C - C \cdot R= & {} L_3 Q(g , C) + L_4 Q(S , C),\\ R \cdot C - C \cdot R= & {} L_5 Q(g , g \wedge S) + L_6 Q(S , g \wedge S), \end{aligned}$$

for some functions $L_1 , \ L_2, \ldots , L_6 : M^{n} \rightarrow \mathbb {R}$. Then we characterize Chen ideal submanifolds $M^n$ of dimension n in $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$, satisfying conditions (i)–(vi). Our main results are presented in Sect. 5. Finally, in Sect. 6 we give proofs of those results.

2 Notations

Let (M , g) be a connected Riemannian $\mathcal {C}^\infty$-manifold of dimension $n \ge 3$ and let $\nabla$ be its Levi-Civita connection, ${\mathcal {X}}(M)$ the Lie algebra of vector fields on M . For vector fields X , Y , Z on M , we define the endomorphism ${\mathcal {R}}(X , Y)$ on ${\mathcal {X}}(M)$ by:

$$\begin{aligned} {\mathcal {R}}(X , Y)Z = \nabla _X\nabla _YZ - \nabla _Y\nabla _XZ - \nabla _{\left[ X , Y\right] }Z . \end{aligned}$$

The Riemann–Christoffel curvature (0 , 4) -tensor R is defined as follows: $R(X , Y , Z , W) = g\left( {\mathcal {R}}\left( X , Y\right) Z ,W\right)$. The Ricci (0 , 2) -tensor S and the Ricci operator ${\mathcal {S}}$ are related by: $S(X , Y) = g\left( {\mathcal {S}}X , Y\right)$. With respect to an orthonormal framefield $\left\{ e_1 , \cdots , e_n\right\}$, one has: $S(X , Y) = \sum _{i=1}^n R(X , e_i , e_i , Y)$. The scalar curvature $\tau$ is given by $\tau = {\mathrm{tr}} (S)$. With respect to an orthonormal framefield $\left\{ e_1 , \ldots , e_n\right\}$, one has: $\tau = \sum _{i=1}^n S(e_i , e_i)$.

Let A be a symmetric (0 , 2) -tensor. To every couple (X , Y) of vector fields on M , one can associate an endomorphism $X \wedge _A Y$ on ${\mathcal {X}}(M)$ by putting:

$$\begin{aligned} \left( X \wedge _A Y\right) (Z) = A(Y , Z)X - A(X , Z)Y . \end{aligned}$$

In particular, when $A = g$,

$$\begin{aligned} \left( X \wedge _g Y\right) (Z) = g(Y , Z)X - g(X , Z)Y . \end{aligned}$$

Let A , B be two symmetric (0 , 2) -tensors on M . Their Kulkarni–Nomizu product $A \wedge B$ is defined on $\left( {\mathcal {X}}(M)\right) ^4$ by:

$$\begin{aligned} \left( A \wedge B\right) (X , Y , Z , W)= & {} A(X , W)B(Y , Z) + A(Y , Z)B(X , W) \\&-A(X , Z)B(Y , W) + A(Y , W)B(X , Z). \end{aligned}$$

In particular, when $A = B = g$, we have the Kulkarni–Nomizu squared $g \wedge g$:

$$\begin{aligned} \left( g \wedge g\right) (X , Y , Z , W) = 2\left[ g(X , W)g(Y , Z) - g(X , Z)g(Y , W)\right] . \end{aligned}$$

We notice that

$$\begin{aligned} \left( g \wedge g\right) (X , Y , Z , W) = 2g\left( (X \wedge _g Y) (Z) , W\right) . \end{aligned}$$

This leads to the (0 , 4) -tensor $G = \left( 1/2\right) \left( g \wedge g\right)$; it is defined as follows:

$$\begin{aligned} G(X , Y , Z , W) = g(X , W)g(Y , Z) - g(X , Z)g(Y , W). \end{aligned}$$

It is well-known that M is of constant curvature c if and only if $R = cG$.

For every vector fields X , Y on M , the endomorphism ${\mathcal {C}}(X , Y)$ on ${\mathcal {X}}(M)$ is given by:

$$\begin{aligned} {\mathcal {C}}(X , Y) = {\mathcal {R}}(X , Y) - \dfrac{1}{n-2}\left[ X\wedge _g \left( {\mathcal {S}}Y\right) + \left( {\mathcal {S}}X\right) \wedge _g Y\right] + \dfrac{\tau }{(n-1)(n-2)}X\wedge _g Y . \end{aligned}$$

The Weyl conformal curvature (0 , 4) -tensor C associated with ${\mathcal {C}}$ is defined by:

$$\begin{aligned} C(X , Y , Z , W) = g\left( {\mathcal {C}}(X , Y)Z , W\right) . \end{aligned}$$

This gives the following relation:

$$\begin{aligned} C = R - \dfrac{1}{n-2}\left( g\wedge S\right) + \dfrac{\tau }{(n-1)(n-2)} G . \end{aligned}$$

(13)

For every vector fields X , Y on M , we consider a skew-symmetric endomorphism ${\mathcal {B}}(X , Y)$ on ${\mathcal {X}}(M)$. We define the (0 , 4) -tensor B associated with ${\mathcal {B}}$ by:

$$\begin{aligned} B(X , Y , Z , W) = g\left( {\mathcal {B}}(X , Y)Z , W\right) . \end{aligned}$$

This tensor B is called a generalized curvature tensor if the following two conditions are fulfilled:

$$\begin{aligned}&B\left( X_1 , X_2 , X_3 , X_4\right) = B\left( X_3 , X_4 , X_1 , X_2\right) ; \\&\quad B\left( X_1 , X_2 , X_3 , X_4\right) + B\left( X_2 , X_3 , X_1 , X_4\right) + B(X_3 , X_1 , X_2 , X_4) = 0. \end{aligned}$$

Now let us extend the endomorphism ${\mathcal {B}}(X , Y)$ to a derivation ${\mathcal {B}}(X , Y)\cdot$ of the algebra of tensor fields on M , assuming that it commutes with contractions and ${\mathcal {B}}(X , Y)\cdot f = 0$ for any smooth real-valued function f on M . Furthermore consider a (0 , k) -tensor T , for $k \ge 1$. We define the $(0 , k + 2)$-tensor $B \cdot T$ by putting:

$$\begin{aligned} \left( B \cdot T\right) \left( X_1 , \ldots , X_k ; X , Y\right)= & {} - T\left( {\mathcal {B}}(X , Y)X_1 , \ldots , X_k\right) \\&- T\left( X_1 , {\mathcal {B}}(X , Y)X_2 , \ \ldots , X_k\right) \\&- \cdots - T\left( X_1 , X_2, \ \ldots , {\mathcal {B}}(X , Y)X_k\right) . \end{aligned}$$

Substituting ${\mathcal {B}} = {\mathcal {R}}$ or ${\mathcal {B}} = {\mathcal {C}}$, and $T = C$ or $T = R$ in the above formulas, we get the tensors: $R \cdot R$, $C \cdot C$, $R \cdot C$ and $C \cdot R$. The two latest lead to the difference tensor $R \cdot C - C \cdot R$. Further, let A be a symmetric (0 , 2) -tensor. Denote by ${\mathcal {A}}$ the endomorphism associated with A by: $g\left( {\mathcal {A}}X , Y\right) = A(X , Y)$. Now we consider a (0 , k) -tensor T , for $k \ge 2$. The Tachibana tensor of A and T (or for short, the Tachibana tensor) Q(A , T) is defined on $\left( {\mathcal {X}}(M)\right) ^k \times \left( {\mathcal {X}}(M)\right) ^2$ by:

$$\begin{aligned} Q(A , T)\left( X_1 , \ldots , X_k ; X , Y\right)= & {} \left( \left( X \wedge _A Y\right) \cdot T\right) \left( X_1 , \ \ldots , X_k\right) \\= & {} - T\left( \left( X \wedge _A Y\right) X_1 , \ \ldots , X_k\right) \\&- T\left( X_1 , \left( X \wedge _A Y\right) X_2 , \ \ldots , X_k\right) \\&- \cdots - T\left( X_1 , X_2 , \ \ldots , \left( X \wedge _A Y\right) X_k\right) . \end{aligned}$$

Substituting $A = g$ or $A = S$, and $T = C$ or $T = R$ or $T = g \wedge S$ in the above formulas, we get one of the following (0 , 6)-Tachibana tensors [33] which may not vanish identically: Q(g , C), Q(g , R), $Q(g , g \wedge S)$, Q(S , C), Q(S , R) and $Q(S , g \wedge S)$. We also have the following identity (e.g. see [26]):

$$\begin{aligned} (n - 2)\left( R \cdot C - C \cdot R\right) = Q\left( S - \dfrac{\tau }{n-1}g , R\right) - g \wedge \left( R \cdot S\right) + P \end{aligned}$$

where the (0 , 6) -tensor P is defined by:

$$\begin{aligned} P\left( X_1 , X_2 , X_3 , X_4 ; X , Y\right)= & {} g(X , X_1)R\left( {\mathcal {S}}(Y), X_2 , X_3 , X_4\right) \\&-\,g(Y , X_1)R\left( {\mathcal {S}}(X), X_2 , X_3 , X_4\right) \\&+ \, g(X , X_2)R\left( X_1 , {\mathcal {S}}(Y), X_3 , X_4\right) \\&-\, g(Y , X_2)R\left( X_1 , {\mathcal {S}}(X), X_3 , X_4\right) \\&+\, g(X , X_3)R\left( X_1 , X_2 , {\mathcal {S}}(Y), X_4\right) \\&-\,g(Y , X_3)R\left( X_1 , X_2 , {\mathcal {S}}(X), X_4\right) \\&+\, g(X , X_4)R\left( X_1 , X_2 , X_3 , {\mathcal {S}}(Y)\right) \\&-\,g(Y , X_4)R\left( X_1 , X_2 , X_3 , {\mathcal {S}}(X)\right) . \end{aligned}$$

It is well-known that the conharmonic curvature tensor conh(R) of a semi-Riemannian manifold (M, g), $n \ge 4$, is defined by

$$\begin{aligned} conh(R)= & {} R - \frac{1}{n-2}\, g \wedge S . \end{aligned}$$

Evidently, conh(R) is a generalized curvature tensor. In addition (13) yields

$$\begin{aligned} conh(R)= & {} C - \frac{\tau }{(n-1)(n-2)}\, G . \end{aligned}$$

(14)

It is clear that

$$\begin{aligned} conh(R) = 0 \iff \left( C = 0 \ \text {and} \ \tau = 0 \right) . \end{aligned}$$

We also have

Proposition 1

[25] For any semi-Riemannian manifold (M, g), $n \ge 4$, the following identities hold good:

$$\begin{aligned} \left\{ \begin{array}{l} \displaystyle conh(R) \cdot S = \displaystyle C \cdot S - \frac{ \tau }{(n-2)(n-1)}\, Q( g, S); \\ \displaystyle R \cdot conh(R) = R \cdot C ; \\ \displaystyle conh(R) \cdot R = C \cdot R - \frac{ \tau }{(n-1)(n-2)}\, Q(g,R) ; \\ \displaystyle conh(R) \cdot conh(R) = C \cdot C - \frac{ \tau }{(n-1(n-2)}\, Q(g, C) . \end{array} \right. \end{aligned}$$

(15)

$\square$

Using the above-presented relations, we get immediately

$$\begin{aligned} \left\{ \begin{aligned} R \cdot conh(R) - conh(R) \cdot R&= R \cdot C - C \cdot R + \frac{ \tau }{(n-1)(n-2)}\, Q(g,R) ,\\ C \cdot conh(R) - conh(R) \cdot C&= C \cdot \left( conh(R) + \frac{\tau }{(n-1)(n-2)}\, G \right) - \left( C - \frac{\tau }{(n-1)(n-2)}\, G \right) \cdot C \\&= \frac{ \tau }{(n-1)(n-2)}\, Q(g,C). \end{aligned} \right. \end{aligned}$$

(16)

We mention that quasi-Einstein manifolds satisfying some curvature conditions were investigated by several authors (e.g. see [18, 26, 27, 38, 42, 51, 65, 70], and references therein). In particular, [65] contains results on quasi-Einstein manifolds satisfying curvature conditions involving the conharmonic tensor conh(R).

We will use the following

Proposition 2

[21, Proposition 4.1], [37, Lemma 3.4] Let (M, g), $n \ge 3$, be a semi-Riemannian manifold. Let a non-zero symmetric (0, 2)-tensor A and a generalized curvature tensor B, defined at $p \in M$, satisfy at this point $Q(A,B) = 0$. In addition, let Y be a vector at p such that the scalar $\rho = w(Y)$ is non-zero, where w is a covector defined by $w(X) = A(X,Y)$, $X \in T_{p}(M)$. Then we have:

(i) $A - \rho \, w \otimes w \ne 0$ and $B = \lambda \, A \wedge A$, $\lambda \in {\mathbb {R}}$, or (ii) $A = \rho \, w \otimes w$ and

$$\begin{aligned}&w(X)\, B(Y,Z, X_{1}, X_{2}) + w(Y)\, B(Z,X, X_{1}, X_{2}) \nonumber \\&\quad + w(Z)\, B(X,Y, X_{1}, X_{2}) \ =\ 0,\ \ X,Y,Z,X_{1},X_{2} \in T_{p}M. \end{aligned}$$

(17)

Moreover, in both cases the following condition holds at p :

$$\begin{aligned} B \cdot B= & {} Q\left( \mathrm{Ric}(B),B\right) , \end{aligned}$$

(18)

where $\mathrm{Ric}(B)$ is the Ricci tensor of B . $\square$

As an immediate consequence of Proposition 2, using the definition of the tensors R and Q(g, R), resp., C and Q(g, C), we can easily check that $Q(g,R) = 0$ at a point of a manifold (M, g), $n \ge 4$, if and only if $R = \left( \tau /\left( (n-1)n\right) \right) \, G$ at this point, resp., $Q(g,C) = 0$ at a point of M if and only if $C = 0$ at this point. On another hand, it is also easy to check that $Q(g, g \wedge S)$ vanishes at a point of a manifold (M, g), $n \ge 4$, if and only if S is proportional to g, i.e. $S = \left( \tau /n\right) \, g$ holds at this point.

It is clear that $Q(S, g \wedge S)$ vanishes at all points at which $S = \left( \tau /n\right) \, g$. Proposition 1.1, Lemma 3.1 of [49] and (10) lead to the following: $Q(S, g \wedge S)$ vanishes at a point of ${\mathcal {U}}_{S} \subset M$ of a manifold (M, g), $n \ge 4$, if and only if $\text{ rank }\, S = 1$ at this point.

With respect to the above-presented material, we restrict our investigation to the set ${\mathcal {U}} = {\mathcal {U}}_{S} \cap {\mathcal {U}}_{C} \subset M$ of a manifold (M, g) , $n \ge 4$. Further, if Q(S, C) is a zero tensor on ${\mathcal {U}} \subset M$ then $R \cdot R = \left( \tau /(n-1)\right) \, Q(g,R)$ (and in consequence $R \cdot C = \left( \tau /(n-1)\right) \, Q(g,C)$) hold on ${\mathcal {U}}$ [36, Theorem 3.1]. Moreover, if $n=4$ then $R \cdot R = 0$ and $\tau = 0$ hold on ${\mathcal {U}}$ [40, Theorem 3.1]. If Q(S, R) is a zero tensor on ${\mathcal {U}} \subset M$ then $R \cdot R = 0$ hold on ${\mathcal {U}}$ [21, Theorem 4.1].

3 On Chen Ideal Submanifolds

3.1 Introduction

Let M be a submanifold of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 2$, $m \ge 1$. Let g be the Riemannian metric induced on M from the standard metric on ${\mathbb {E}}^{n+m}$, $\nabla$ the corresponding Levi-Civita connection on M, and R, S, $\tau$ respectively the Riemann–Christoffel curvature tensor, the Ricci tensor and the scalar curvature of M.

For the scalar curvature $\tau$ of (M , g) we use the calibration

$$\begin{aligned} \tau (p) := \displaystyle \sum _{i < j} K\left( p , e_i(p) \wedge e_j(p)\right) , \end{aligned}$$

where $K\left( p , \pi \right)$ denotes the Riemannian sectional curvature of (M , g) at the point p for a plane section $\pi$ in the tangent space $T_p M$. For each point p in M, considering the number

$$\begin{aligned} \left( \mathrm{inf}\, K\right) (p) : = \mathrm{inf} \left\{ K(p , \pi ) \vert \pi \ \text {is a plane section in} \ T_p M\right\} , \end{aligned}$$

B.-Y. Chen (see [6, 9]) introduced the $\delta (2)$-curvature by

$$\begin{aligned} \left( \delta (2)\right) (p) = \delta (p) := \tau (p) - \left( \mathrm{inf}\, K\right) (p). \end{aligned}$$

This $\delta (2)$-, for short, $\delta$ -curvature of Chen thus is a well-defined real function on M which clearly is a Riemannian invariant of (M , g). From [9] (see also [6, 8, 16]), we have the following basic result which, in particular, answered a question raised by S.S. Chern [17] long before, concerning intrinsic obstructions on Riemannian manifolds in view of minimal immersibility in Euclidean spaces.

Theorem 1

[6] For any submanifold M of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 2$, $m \ge 1$,

$$\begin{aligned} \delta \le \dfrac{n^2(n-2)}{2(n-1)} H^2 , \quad \mathrm {(*)} \end{aligned}$$

and in (*) equality holds at a point $p \in M$ if and only if, with respect to some suitable adapted orthonormal frame $\left\{ e_i , \xi _\alpha \right\}$ around p on M in ${\mathbb {E}}^{n+m}$, the shape operators are given by

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} b &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} z &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} z \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta > 1 , \end{aligned}$$

where $z = a + b$ and $(\inf K) = ab - \displaystyle \sum _{\beta >1} \left( c^2_\beta + d^2_\beta \right) : M \rightarrow \mathbb {R}$. $\square$

Evidently, if $m = 1$ then $\inf K = ab$.

With respect to the above theorem, one has the following definition (see [6, 8, 32, 43, 69]).

Definition 1

Let M be a submanifold of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 2$, $m \ge 1$. It is called a Chen ideal submanifold if, at each of its points, the Chen’s basic inequality (*) in the Theorem 1 is actually an equality.

Let M be a Chen ideal submanifold of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. We use the notations as in Theorem 1. The Riemann–Christoffel curvature tensor R satisfies:

$$\begin{aligned} \left\{ \begin{aligned} R\left( e_1 , e_2 , e_2 , e_1\right)&= \inf K = ab - \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) ; \\ R\left( e_1, e_i, e_i, e_1\right)&= az \quad \mathrm{for} \quad i \ge 3 ; \\ R\left( e_2, e_i, e_i, e_2\right)&= bz \quad \mathrm{for} \quad i \ge 3 ; \\ R\left( e_i, e_j, e_j, e_i\right)&= z^2 \quad \mathrm{for} \quad 3 \le i < j \le n . \\ \end{aligned} \right. \end{aligned}$$

(19)

The other values of $R\left( e_u , e_v , e_w , e_t\right)$ are null. The Ricci tensor S satisfies:

$$\begin{aligned} \left\{ \begin{aligned} S\left( e_1 , e_1\right)&= \inf K + (n-2)az ; \\ S\left( e_2 , e_2\right)&= \inf K + (n-2)bz ; \\ S\left( e_i , e_i\right)&= (n-2)z^2 \quad \mathrm{for} \quad 3 \le i \le n ; \\ S\left( e_u , e_v\right)&= 0 \quad \mathrm{for} \quad 1 \le u < v \le n. \\ \end{aligned} \right. \end{aligned}$$

(20)

The scalar curvature $\tau$ is given by:

$$\begin{aligned} \tau = \sum _{i=1}^n \mathrm{S}(e_i , e_i) = 2 \inf K + (n-1)(n-2)z^2. \end{aligned}$$

(21)

The Weyl conformal curvature tensor C is determined by the following relations:

$$\begin{aligned} \left\{ \begin{aligned} C\left( e_1 , e_2 , e_2 , e_1\right)&= \dfrac{(n - 3)(\inf K)}{n - 1} ; \\ C\left( e_1 , e_i , e_i , e_1\right)&= -\dfrac{(n - 3)(\inf K)}{(n - 1)(n -2)} \quad {\mathrm{for}} \quad i \ge 3 ; \\ C\left( e_2 , e_i , e_i , e_2\right)&= -\dfrac{(n - 3)(\inf K)}{(n - 1)(n -2)} \quad {\mathrm{for}} \quad i \ge 3 ; \\ C\left( e_i , e_j , e_j , e_i\right)&= \dfrac{2(\inf K)}{(n - 1)(n -2)} \quad {\mathrm{for}} \quad 3 \le i < j \le n . \\ \end{aligned} \right. \end{aligned}$$

(22)

From (22) it follows (cf. [32, 43], Theorem F) that every Chen ideal submanifold M of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$, has a pseudo-symmetric Weyl conformal curvature C , i.e. it satisfies the identity:

$$\begin{aligned} C \cdot C= & {} L_C\, Q(g,C) ,\ \ L_C \ =\ - \dfrac{(n-3)(\inf K)}{(n-1)(n-2)} . \end{aligned}$$

(23)

Very recently semi-Riemannian manifolds satisfying (23) were investigated in [24].

It is known that at every point of a hypersurface N in a space forms $\widetilde{N}^{n+1}(c)$, $n \ge 4$, the tensors $R \cdot R - Q(S,R)$ and Q(g, C) are linearly dependent. Precisely, we have on N [44]

$$\begin{aligned} R \cdot R - Q(S,R) = - (n-2) c \, Q(g,C) . \end{aligned}$$

Thus, in particular, $R \cdot R - Q(S,R) = 0$ on every Chen ideal hypersurface N in ${\mathbb {E}}^{n+1}$, $n \ge 4$.

Now let us compute the difference $R \cdot R - Q(S,R)$ on the Chen ideal submanifold M of codimension m in ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. With respect to the notations in Theorem 1 and from the equalities (19), (20) and (22), we can prove the following.

Theorem 2

The identity

$$\begin{aligned} R \cdot R - Q(S,R) = \dfrac{ab - \inf K}{\inf K} (n-2)z^2\, Q(g,C) \end{aligned}$$

(24)

holds on the subset ${\mathcal {U}}_{C}$ (see Sect. 1) of every Chen ideal submanifold M of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. In addition, at each point $p \in M$ where C vanishes ($\inf K = 0$), the following equalities hold:

$$\begin{aligned} \left\{ \begin{aligned} \left( R \cdot R - Q(S,R)\right) \left( e_1 , e_2 , Z , W; e_1 , e_i\right)&= (n-3)abz^2\left<\left( e_2\wedge _g e_i\right) (Z), W\right>, \\ \left( R \cdot R - Q(S,R)\right) \left( e_1 , e_j , Z , W; e_1 , e_i\right)&= -abz^2\left<\left( e_j \wedge _g e_i\right) (Z) ,W\right>, \\ \left( R \cdot R - Q(S,R)\right) \left( e_1 , e_2 , Z , W; e_2 , e_i\right)&= -(n-3)abz^2\left<\left( e_1\wedge _g e_i\right) (Z), W\right>, \\ \left( R \cdot R - Q(S,R)\right) \left( e_2 , e_j , Z , W; e_2 , e_i\right)&= -abz^2\left<\left( e_j \wedge _g e_i\right) (Z) ,W\right> \\ \end{aligned} \right. \qquad \end{aligned}$$

(25)

the other values of $\left( R \cdot R - Q(S,R)\right) \left( e_u , e_v, Z , W; e_w , e_t\right)$ being null. $\square$

As it was proved in [31], every warped product manifold $\overline{M} \times _{F} \widetilde{N}$ of a 2-dimensional base manifold $(\overline{M},\overline{g})$ and an $(n-2)$-dimensional fibre, which is a space of constant curvature $(\widetilde{N},\widetilde{g})$, $n\ge 4$, with the warping function F, satisfies

$$\begin{aligned} C \cdot C = - \frac{(n-3) \rho }{(n-2)(n-1)} \, Q(g,C) ,\ \ \ \ \rho = \frac{ \overline{\tau }}{2} + \frac{\widetilde{\tau }}{(n-3) (n-2) F } + \frac{\Delta F}{2 F} - \frac{\Delta _1 F}{2 F^{2}}, \end{aligned}$$

(26)

where $\Delta F = g^{ab} \nabla _{b} F_{a}$, $\Delta _1 F = g^{ab} F_{a} F_{b}$, and $\overline{\tau }$, $\widetilde{\tau }$ are the scalar curvatures of the base and the fibre, respectively, see to [31] for details. According to [19], every non-trivial and non-minimal Chen ideal submanifold M of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$ is isometric to an open subset of a warped product $\overline{M} \times _{F} {\mathbb {S}}^{n-2}$ of a 2-dimensional base manifold $(\overline{M},\overline{g})$ and an $(n-2)$-dimensional unit sphere ${\mathbb {S}}^{n-2}$, where the warping function F is a solution of some second-order quasilinear elliptical partial differential equation in the plane. Thus we see that (26) holds on M. Furthermore, from (23) and (26) it follows that $\inf K$ is expressed on M by

$$\begin{aligned} \inf K= & {} \frac{ \overline{\tau } }{2} + \frac{ 1 }{ F } + \frac{\Delta F}{2 F} - \frac{\Delta _1 F}{2 F^{2}}. \end{aligned}$$

Since the scalar curvature $\tau$ of M is given by (21) and satisfies (e.g. see [31]):

$$\begin{aligned} \tau= & {} \overline{\tau } + \frac{ (n-3)(n-2) }{ F } - \frac{(n-2) \Delta F}{ F} - \frac{(n-2) (n-5) \Delta _1 F}{4 F^{2}}, \end{aligned}$$

we get:

$$\begin{aligned} (n-2) z^{2}= & {} \frac{ n-4 }{ F } - \frac{\Delta F}{ F} - \frac{ (n-6) \Delta _1 F}{4 F^{2}} . \end{aligned}$$

3.2 On Pseudo-symmetric Chen Ideal Submanifolds

Semi-symmetric spaces have been investigated first by E. Cartan in 1946 [5]. In 1982 [64], Z.I. Szabó established the classification of semi-symmetric spaces. In 1997 [48], F. Dillen and two of the present authors classified all Chen ideal submanifolds which are semi-symmetric.

Theorem 3

[48] A Chen ideal submanifold M of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 3$, $m \ge 1$, is semi-symmetric if and only if M is minimal (in which case M is $(n-2)$-ruled)) or M is a round hypercone in some totally geodesic subspace ${\mathbb {E}}^{n+1}$ of ${\mathbb {E}}^{n+m}$. $\square$

In [32] and [43], Chen ideal pseudo-symmetric submanifolds were classified.

Theorem 4

[32, 43] A Chen ideal submanifold M of codimension m in ${\mathbb {E}}^{n+m}$ ($n \ge 3$, $n \ge 1$) is pseudo-symmetric if and only if:

• (i) either M is semi-symmetric (see Theorem 3),

• (ii) or at every point p of M where $R \cdot R \ne 0$, the 2D normal section $\Sigma ^2_{\tilde{\pi }} \subset {\mathbb {E}}^{2+m}$ of $M^n$ at p in the direction of the tangent plane ${\tilde{\pi }} \subset T_p M^n$ for which the sectional curvature function $K(p , \pi )$ at p attains its minimal value $\left( \inf K\right) (p)$ is pseudo-umbilical at p, or equivalently, if p is a spherical point of the projection $\overline{\Sigma }^2_{\tilde{\pi }} \subset {\mathbb {E}}^{3}$ of this 2D normal section $\overline{\Sigma }^2_{\tilde{\pi }}$ on the space ${\mathbb {E}}^{3}$ spanned by ${\tilde{\pi }}$ and the mean curvature vector $\mathop {H}\limits ^{\rightarrow }(p)$ of $M^n$ in ${\mathbb {E}}^{n+m}$ at p (and in this case $L_R =\left( n^2/\left( 2(n-1)^2\right) \right)$, where H is the mean curvature of $M^n$ in ${\mathbb {E}}^{n+m}$). $\square$

4 On Chen Ideal Submanifolds and Roter Manifolds

A Riemannian manifold (M, g) of dimension n, $n \ge 4$, is said to be a Roter manifold or a Roter space (e.g. see [26, 32] and references therein) if

$$\begin{aligned} R= & {} \frac{\phi }{2}\, S \wedge S + \mu \, g \wedge S + \eta \, G \end{aligned}$$

(27)

holds on ${\mathcal {U}} = {\mathcal {U}}_{S} \cap {\mathcal {U}}_{C} \subset M$, where $\phi$, $\mu$ and $\eta$ are some functions on this set.

According to [32], from a geometric point of view, the pseudo-symmetric Riemannian manifolds can be seen as the most natural symmetric spaces after the real space forms, i.e. the spaces of constant Riemannian sectional curvature. From an algebraic point of view, the Roter manifolds can be seen as the Riemannian manifolds whose Riemann–Christoffel curvature tensor R has the most simple expression after the real space forms, the latter ones being characterisable as the Riemannian spaces $(M^{n}, g)$ for which the (0, 4)-tensor R is proportional to the Nomizu–Kulkarni square of their (0, 2)-metric tensor g. As it was stated in [32], every Chen ideal submanifold M of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$, is a Roter manifold if and only if it is pseudo-symmetric.

As we already mentioned, in this paper we investigate Chen ideal submanifolds (in Euclidean spaces) satisfying some curvature conditions of pseudo-symmetry type. We prove that those submanifolds are pseudo-symmetric and, as a consequence, Roter manifolds too [32].

Using (27), (10) and Theorem 6.7 of [26] we can easy check that the difference tensor $R\cdot C - C \cdot R$ of every Roter manifold (M, g), $n \ge 4$, can be expressed on ${\mathcal {U}} = {\mathcal {U}}_{S} \cap {\mathcal {U}}_{C}$ as a linear combination of the tensors $Q(g, g \wedge S)$ and $Q(S, g \wedge S)$, precisely on this set we have

$$\begin{aligned} R\cdot C - C \cdot R= & {} Q\left( \left( \dfrac{1 }{n-2} - \mu - \dfrac{\phi \tau }{n-1}\right) \, S + \left( \dfrac{\mu \tau }{n-1} + \eta \right) \, g , g \wedge S\right) . \end{aligned}$$

(28)

We note that if (M, g), $n \ge 4$, is a Roter manifold then at every point of ${\mathcal {U}} \subset M$ we must have $\text{ rank }\, (S - \alpha \, g) > 1$, for any real number $\alpha \in \mathbb {R}$.

5 Main Results

Now we give our main results about Chen ideal submanifolds in Euclidean spaces whose difference tensor $R \cdot C - C \cdot R$ can be expressed in terms of some of the Tachibana tensors Q(g , R), Q(S , R), Q(g , C), Q(S , C), $Q(g , g\wedge S)$, $Q(S , g\wedge S)$.

Theorem 5

Let M be a non-conformally flat Chen ideal submanifold of codimension m in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exist two real-valued functions $L_1$, $L_2$ on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L_1 Q(g , R) + L_2 Q(S , R), \end{aligned}$$

if and only if there exists an orthonormal tangent framefield $\left\{ e_1, \ldots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

$$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m, \end{aligned}$$

are given by:

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} \epsilon a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} (1 + \epsilon )a &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} (1 + \epsilon )a \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

where $\epsilon = \pm 1$, a, $c_\beta$ , $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that

$$\begin{aligned} \displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = \epsilon a^2 - \inf K, \ 2\inf K- (1+\epsilon )a^2 \ne 0, \ \inf K \ne (n-2)(1 + \epsilon )a^2 \end{aligned}$$

and

$$\begin{aligned} L_1= & {} \dfrac{(n-3)\inf K - (n - 2)(1+\epsilon )a^2}{(n-1)(n-2)} \dfrac{2\inf K}{2\inf K- (1+\epsilon )a^2} , \\&(1+\epsilon )a^2\left[ L_2 - \dfrac{1}{(n-2)}\dfrac{\inf K}{2\inf K- (1+\epsilon )a^2}\right] = 0. \end{aligned}$$

In this case, M is a Roter space. In addition one has one of the following two situations.

• (i) Either $\epsilon = -1$ and M is a semi-symmetric and minimal submanifold (see Theorem 3) such that

  $$\begin{aligned} R \cdot C - C \cdot R = \dfrac{(n-3)\inf K}{(n-1)(n-2)} Q(g , R). \end{aligned}$$

• (ii) Or $\epsilon = +1$ and M is a properly pseudo-symmetric and non-minimal submanifold (see Theorem 4) such that

  $$\begin{aligned} R \cdot C - C \cdot R= & {} \dfrac{(n-3)\inf K - 2(n - 2)a^2}{(n-1)(n-2)}\dfrac{\inf K}{\inf K- a^2} Q(g , R)\\&\quad + \dfrac{1}{2(n-2)}\dfrac{\inf K}{\inf K- a^2} Q(S , R) . \end{aligned}$$

  $\square$

Corollary 1

Let M be a non-conformally flat Chen ideal submanifold of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exists a real-valued function L on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L Q(g, R) \end{aligned}$$

if and only if M is minimal and there exists an orthonormal tangent framefield $\left\{ e_1, \ldots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1 , \ldots , \xi _m\right\}$ such that the shape operators

$$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m, \end{aligned}$$

are given by:

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} -a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

where a, $c_\beta$, $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that

$$\begin{aligned} \displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = -a^2 - \inf K \end{aligned}$$

and

$$\begin{aligned} L = \dfrac{(n-3)(\inf K)}{(n-1)(n-2)} . \end{aligned}$$

In this case, M is semi-symmetric (see Theorem 3).$\square$

Corollary 2

Let M be a Chen ideal submanifold of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. Then the difference tensor $R \cdot C - C \cdot R$ and the Tachibana tensor Q(S , R) are linearly dependent if and only if M is conformally flat ($\inf K = 0$). $\square$

Theorem 6

Let M be a non-conformally flat Chen ideal submanifold of dimension n in the Euclidean space ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exists two real-valued functions $L_3$, $L_4$ on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L_3 Q(g , C) + L_4 Q(S , C) , \end{aligned}$$

if and only if there exists an orthonormal tangent framefield $\left\{ e_1, \ldots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

$$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m, \end{aligned}$$

are given by:

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} \epsilon a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} (1 + \epsilon )a &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} (1 + \epsilon )a \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

where $\epsilon = \pm 1$, a , $c_\beta$ , $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that

$$\begin{aligned} \displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = \epsilon a^2 - \inf K \end{aligned}$$

and

$$\begin{aligned} L_3 = - \dfrac{2\inf K + 2(n-1)(n-2)(1 + \epsilon )a^2}{n-1} , \quad L_4 = - 1 . \end{aligned}$$

In this case, M is a Roter space. In addition one has one of the following two situations.

• (i) Either $\epsilon = -1$ and M is a semi-symmetric and minimal submanifold (see Theorem 3) such that

  $$\begin{aligned} R \cdot C - C \cdot R = \dfrac{2\inf K}{n-1} Q(g , C) - Q(S , C). \end{aligned}$$

• (ii) Or $\epsilon = +1$ and M is a properly pseudo-symmetric and non-minimal submanifold (see Theorem 4) such that

  $$\begin{aligned} R \cdot C - C \cdot R = - \dfrac{2\inf K + 4(n-1)(n-2)a^2}{n-1} Q(g , C) - Q(S , C) . \end{aligned}$$

  $\square$

Corollary 3

Let M be a Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. Then the difference tensor $R \cdot C - C \cdot R$ and the Tachibana tensor Q(g , C) are linearly dependent if and only if M is conformally flat. $\square$

Corollary 4

Let M be a non-conformally flat Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exists a real-valued function L on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L Q(S , C) , \end{aligned}$$

if and only if M is not minimal, and there exists an orthonormal tangent framefield $\left\{ e_1 , \ldots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

$$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m, \end{aligned}$$

are given by:

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 2a &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 2a \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

where a , $c_\beta$ , $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that $\displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = \left( 2n^2 - 6n + 5\right) a^2 > 0$ and

$$\begin{aligned} L = - 1 . \end{aligned}$$

In this case, $M^n$ is properly pseudo-symmetric (see Theorem 4). $\square$

Theorem 7

Let M be a non-conformally flat Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exists two real-valued functions $L_5$, $L_6$ on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L_5 Q(g, g \wedge S) + L_6 Q(S , g \wedge S) , \end{aligned}$$

if and only if there exists an orthonormal tangent framefield $\left\{ e_1, \ldots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

$$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m, \end{aligned}$$

are given by:

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} \epsilon a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} (1 + \epsilon )a &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} (1 + \epsilon )a \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

where $\epsilon = \pm 1$, a , $c_\beta$, $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that

$$\begin{aligned} \displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = \epsilon a^2 - \inf K , \ \inf K \ne (n-2)(1 + \epsilon )a^2 \end{aligned}$$

and moreover

$$\begin{aligned} \begin{aligned} L_5&= - \dfrac{2(n-2)(1 + \epsilon )a^2}{n-1}\dfrac{\inf K\left[ \inf K + (n-2)(1 + \epsilon )a^2\right] }{\left[ \inf K - (n-2)(1 + \epsilon )a^2\right] ^2} ,\\ L_6&= - \dfrac{1}{(n-1)(n-2)}\dfrac{\inf K\left[ (n-3)\inf K + (n-1)(n-2)(1 + \epsilon )a^2\right] }{\left[ \inf K - (n-2)(1 + \epsilon )a^2\right] ^2} .\\ \end{aligned} \end{aligned}$$

In this case, M is a Roter space. In addition one has one of the following two situations.

• (i) Either $\epsilon = -1$ and M is a semi-symmetric and minimal submanifold (see Theorem 3) such that

  $$\begin{aligned} R \cdot C - C \cdot R = - \dfrac{n-3}{(n-1)(n-2)} Q(S, g \wedge S). \end{aligned}$$

• (ii) Or $\epsilon = +1$ and M is a properly pseudo-symmetric and non-minimal submanifold (see Theorem 4) such that

  $$\begin{aligned}&R \cdot C - C \cdot R = - \dfrac{4(n-2)a^2}{n-1}\dfrac{\inf K\left[ \inf K + 2(n-2)a^2\right] }{\left[ \inf K - 2(n-2)a^2\right] ^2} Q(g , g \wedge S) \\&\quad - \dfrac{1}{(n-1)(n-2)}\dfrac{\inf K\left[ (n-3)\inf K + 2(n-1)(n-2)a^2\right] }{\left[ \inf K - 2(n-2)a^2\right] ^2} Q(S , g \wedge S) . \end{aligned}$$

  $\square$

Corollary 5

Let M be a non-conformally Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exists a real-valued function L on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L Q(g, g \wedge S), \end{aligned}$$

if and only if M is not minimal, and there exists an orthonormal tangent framefield $\left\{ e_1 , \ldots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

$$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m, \end{aligned}$$

are given by:

$$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 2a &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 2a \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

where a , $c_\beta$ , $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that

$$\begin{aligned} \displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = \dfrac{2n^2 - 5n + 1}{n-1} a^2 > 0 \end{aligned}$$

and

$$\begin{aligned} L = - \dfrac{2a^2}{n-2} . \end{aligned}$$

In this case, $M^n$ is a properly pseudo-symmetric manifold (see Theorem 4). $\square$

Corollary 6

Let M be a non-conformally flat Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. Then there exists a real-valued function L on M such that

$$\begin{aligned} R \cdot C - C \cdot R = L Q(S , g \wedge S), \end{aligned}$$

if and only if one has one of the two cases which follow.

• (i) Either M is minimal, and there exists an orthonormal tangent framefield $\left\{ e_1 , \cdots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

  $$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m , \end{aligned}$$

  are given by:

  $$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} -a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix}, \ \beta \ge 2 , \end{aligned}$$

  where a , $c_\beta$ , $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that

  $$\begin{aligned} \displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = - a^2 - \inf K \end{aligned}$$

  and

  $$\begin{aligned} L = - \dfrac{n-3}{(n-1)(n-2)} . \end{aligned}$$

• (ii) Or M is not minimal, and there exists an orthonormal tangent framefield $\left\{ e_1 , \cdots , e_n\right\}$ and an orthonormal normal framefield $\left\{ \xi _1, \ldots , \xi _m\right\}$ such that the shape operators

  $$\begin{aligned} A_\alpha := A_{\xi _\alpha }, \quad 1 \le \alpha \le m \end{aligned}$$

  are given by:

  $$\begin{aligned} A_1 = \begin{pmatrix} a &{} 0 &{} 0 &{} \cdots &{} 0 \\ 0 &{} a &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 2a &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 2a \\ \end{pmatrix} , \quad A_\beta = \begin{pmatrix} c_\beta &{} d_\beta &{} 0 &{} \cdots &{} 0 \\ d_\beta &{} -c_\beta &{} 0 &{} \cdots &{} 0 \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ 0 &{} 0 &{} 0 &{} \cdots &{} 0 \\ \end{pmatrix} , \ \beta \ge 2 , \end{aligned}$$

  where a , $c_\beta$ , $d_\beta$ (for $2 \le \beta \le m$) are real functions on M such that $\displaystyle \sum _{\beta = 2}^m \left( c^2_\beta + d^2_\beta \right) = (2n - 3) a^2$ and

  $$\begin{aligned} L = \dfrac{1}{2\left( n-1\right) \left( n-2\right) } . \end{aligned}$$

In the first case, M is a semi-symmetric manifold (see Theorem 3). In the second case, M is a properly pseudo-symmetric manifold (see Theorem 4). $\square$

Corollary 7

Let M be a non-conformally Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. If M is minimal, then:

$$\begin{aligned} R \cdot C - C \cdot R = - \dfrac{n-3}{(n-1)(n-2)} Q(S , g \wedge S). \end{aligned}$$

$\square$

Corollary 8

A Chen ideal submanifold M of dimension $n \ge 4$ in the Euclidean space $\mathbb {E}^{n+m}$ satisfies the curvature condition

$$\begin{aligned} R \cdot C - C \cdot R = 0 \end{aligned}$$

if and only if M is conformally flat. $\square$

With respect to (28), as an immediate consequence of Theorem 7, for Chen ideal and Roter submanifolds, we express the difference tensor $R \cdot C - C \cdot R$ as a linear combination of the Tachibana tensors $Q(g , g \wedge S)$, $Q(S , g \wedge S)$, in terms of $\inf K$ (the infimum of all of its sectional curvatures) and the scalar curvature $\tau$.

Corollary 9

Let M be a non-conformally flat Chen ideal submanifold of dimension n in the Euclidean space $\mathbb {E}^{n+m}$, $n \ge 4$, $m \ge 1$. If M is a Roter space, then:

$$\begin{aligned} R \cdot C - C \cdot R= & {} - \dfrac{2\left[ \tau - 2\inf K\right] \left[ \tau + 2(n - 2)\inf K\right] }{(n-1)\left[ \tau - 2n(\inf K)\right] ^2}Q(g , g \wedge S)\\&- \dfrac{2(n - 1)(\inf K)\left[ \tau + 2(n - 4)\inf K\right] }{(n-2)\left[ \tau - 2n(\inf K)\right] ^2}Q(S , g \wedge S) . \end{aligned}$$

$\square$

Finally, in addition to (16), as an immmediate consequence of (23), (15) and (14), we get the following theorem:

Theorem 8

Every Chen ideal submanifold M of codimension m in ${\mathbb {E}}^{n+m}$, $n \ge 4$, $m \ge 1$, satisfies:

$$\begin{aligned} conh(R) \cdot conh(R)= & {} - \frac{ \tau + (n-3) (\inf K)}{(n-1)(n-2)}\, Q(g,C)\\= & {} - \frac{ \tau + (n-3) (\inf K)}{(n-1)(n-2)}\, Q(g,conh(R)). \end{aligned}$$

$\square$

We refer to [56] and [62] for further results on Chen ideal submanifolds satisfying curvature conditions involving the conharmonic curvature tensor.

6 Proofs of Main Results

In this section, we consider a Chen ideal submanifold M of codimension m in the Euclidean space ${\mathbb {E}}^{n+m}$. We use the notations as in Theorem 1. To prove all our main results on M , we begin by computing the Tachibana tensor Q(A , T) on M , for any symmetric (0 , 2) -tensor A and for any generalized curvature (0 , 4) -tensor T . We then determine the properties of the difference tensor $R \cdot C - C \cdot R$.

Proposition 3

Let M be a Chen ideal submanifold of codimension m in the Euclidean space $\mathbb {E}^{n+m}$. Consider a symmetric (0 , 2) -tensor A and a generalized curvature (0 , 4) -tensor T defined on M . With respect to the notations of Definition 1, we put

$$\begin{aligned} T_{\alpha \beta \gamma \delta } = T\left( e_\alpha , e_\beta , e_\gamma , e_\delta \right) , \quad A_{\alpha \beta } = A\left( e_\alpha , e_\beta \right) \end{aligned}$$

for any different indices $\alpha$, $\beta$, $\gamma$, $\delta \in \left\{ 1 , \ldots , n\right\}$. Then, for any tangent vector field X , Y on M and for any different indices u , v , s , t , $\beta \in \left\{ 1 , \ldots , n\right\}$, one has the following three identities:

$$\begin{aligned}&Q(A , T)\left( e_u , e_s , X , Y; e_u , e_v\right) \\&\quad = T_{ussu}\left[ A_{uv}\Big <\left( e_s \wedge _g e_u\right) (X) , Z\Big > +\sum _{\alpha =1}^n A_{\alpha v}\Big <\left( e_s \wedge _g e_\alpha \right) (X) , Y\Big >\right] \\&\qquad - T_{vssv}A_{uu}\Big <\left( e_s \wedge _g e_v\right) (X) , Y\Big > + T_{uvvu}A_{su}\Big <\left( e_u \wedge _g e_v\right) (X) , Y\Big > ;\\&\quad Q(A , T)\left( e_u , e_v , X , Y; e_u , e_v\right) = 0 ;\\&\quad Q(A , T)\left( e_{s}, e_{t} , X , Y; e_u , e_v\right) = 0 . \end{aligned}$$

$\square$

Proof of Proposition 3

Let X , Y be two tangent vector fields on M . It is useful to recall here that, for any tangent vector fields U and V on M , one has:

$$\begin{aligned}&Q(A , T)\left( U , V , X , Y; e_u , e_v\right) \\&\quad = - T\left( \left( e_u\wedge _A e_v\right) (U), V , X , Y\right) - T\left( U , \left( e_u\wedge _A e_v\right) (V), X , Y\right) \\&\qquad - T\left( U , V , \left( e_u\wedge _A e_v\right) (X), Y\right) - T\left( U , V , X , \left( e_u\wedge _A e_v\right) (Y)\right) . \end{aligned}$$

Then for any three different indices u , v , s , $t \in \left\{ 1 , \ldots , n\right\}$ and substituting (U , V , X , Y) successively by $\left( e_u , e_s , X , Y\right)$, $\left( e_u , e_v , X , Y\right)$ and $\left( e_s , e_t , X , Y\right)$ in the above formula, we get straightforwardly all the formulas in Proposition 3. $\square$

Proposition 4

For all tangent vector field X , Y , Z , W and for all different indices u , $v \in \left\{ 1, \ldots , n\right\}$,

$$\begin{aligned}&\left( R \cdot C - C \cdot R\right) \left( X , Y , Z , W ; e_u , e_v\right) \\&\quad = \left[ R_{uvvu}Q(g , C) - C_{uvvu}Q(g , R)\right] \left( X , Y , Z , W ; e_u , e_v\right) . \end{aligned}$$

$\square$

Proof of Proposition 4

Let X , Y , Z , W be four tangent vector fields and u , v different indices such that $1 \le u \le n$ and $1 \le v \le n$. On the one hand,

$$\begin{aligned} \begin{aligned} \left( R \cdot C\right) \left( X , Y , Z , W ; e_u , e_v\right)&= R_{uvvu}Q(g , C)\left( X , Y , Z , W ; e_u , e_v\right) ; \\ \left( C \cdot R\right) \left( X , Y , Z , W ; e_u , e_v\right)&= C_{uvvu}Q(g , R)\left( X , Y , Z , W ; e_u , e_v\right) .\\ \end{aligned} \end{aligned}$$

Then the difference gives directly the required equality. $\square$

6.1 Proof of Theorem 5

To deduce Theorem 5, one can use directly the notations of Theorem 1 and the following Lemma 1.

Lemma 1

Let M be a Chen ideal and non-conformally flat submanifold of dimension $n \ge 4$ in the Euclidean space $\mathbb {E}^{n+m}$. There exists two real-valued functions $L_1$, $L_2$ on M such that

$$\begin{aligned} L_1 Q(g , R) + L_2 Q(S , R) = R \cdot C - C \cdot R \end{aligned}$$

if and only if

$$\begin{aligned} \left\{ \begin{aligned}&b = \epsilon a \\&\left( 1 + \epsilon \right) a^2L_1 + 2\left( 1 + \epsilon \right) a^2\left( \inf K\right) L_2 = \dfrac{2\left( 1 + \epsilon \right) a^2\inf K}{n-1}\\&\left( \inf K\right) L_1 + \Big [(2n - 3)\left( 1 + \epsilon \right) a^2\left( \inf K\right) - 2(n - 2)\left( 1 + \epsilon \right) a^4\Big ]\\&\quad L_2 = \dfrac{\left( \inf K\right) \Big [(n - 3)\left( \inf K\right) + (n-1)(n-2)\left( 1 + \epsilon \right) a^2\Big ]}{(n-1)(n-2)}, \\ \end{aligned} \right. \end{aligned}$$

where $\epsilon = \pm 1$.$\square$

Proof of Lemma 1

Using the relations given in Propositions 3 and 4, we obtain straightforwardly the required equivalences. $\square$

6.2 Proof of Theorem 6

To deduce Theorem 6, one can use directly the notations of Theorem 1 and the following Lemma 2.

Lemma 2

Let M be a Chen ideal and non-conformally flat submanifold of dimension $n \ge 4$ in the Euclidean space $\mathbb {E}^{n+m}$. There exists two real-valued functions $L_3$, $L_4$ on M such that

$$\begin{aligned} L_3 Q(g , C) + L_4 Q(S , C) = R \cdot C - C \cdot R \end{aligned}$$

if and only if

$$\begin{aligned} \left\{ \begin{aligned}&b = \epsilon a \\&L_3 + \dfrac{\inf K + (n-2)(2n-3)\left( 1 + \epsilon \right) a^2}{n-1}L_4 = \dfrac{\inf K + (n-2)\left( 1 + \epsilon \right) a^2}{n-1} \\&L_3 + \dfrac{2\inf K + 2(n-2)^2\left( 1 + \epsilon \right) a^2}{n-1}L_4 = \dfrac{2(n-2)\left( 1 + \epsilon \right) a^2}{n-1}, \\ \end{aligned} \right. \end{aligned}$$

where $\epsilon = \pm 1$.$\square$

Proof of Lemma 2

Using the relations given in Propositions 3 and 4, we obtain straightforwardly the required equivalences. $\square$

6.3 Proof of Theorem 7

To deduce Theorem 7, one can use directly the notations of Theorem 1 and the following Lemma 3.

Lemma 3

Let M be a Chen ideal and non-conformally flat submanifold of dimension $n \ge 4$ in the Euclidean space $\mathbb {E}^{n+m}$. There exists two real-valued functions $L_5$, $L_6$ on M such that

$$\begin{aligned} L_5 Q(g , g \wedge S) + L_6 Q(S , g \wedge S) = R \cdot C - C \cdot R \end{aligned}$$

if and only if

$$\begin{aligned} \left\{ \begin{aligned}&b = \epsilon a \\&L_5 - \left[ \inf K + (n-2)\left( 1+\epsilon \right) a^2\right] L_6 = \left( \inf K\right) \dfrac{n-3}{(n-1)(n-2)} \dfrac{\inf K + (n-2)\left( 1 + \epsilon \right) a^2}{\inf K - (n-2)\left( 1 + \epsilon \right) a^2}\\&L_5 - \left[ 2(n-2)\left( 1+\epsilon \right) a^2\right] L_6 = -\dfrac{2a^2\left( 1+\epsilon \right) }{n-1} \dfrac{\inf K}{\inf K - (n-2)\left( 1 + \epsilon \right) a^2}\\ \end{aligned} \right. \end{aligned}$$

where $\epsilon = \pm 1$ .$\square$

Proof of Lemma 3

Using the relations given in Propositions 3 and 4, we obtain straightforwardly the required equivalences. $\square$