Semi-invariant Submanifolds in Metric Geometry of Endomorphisms

Abstract

Attempts have been made to introduce generalization of structured manifolds as the most general Riemannian metric g associated with an endomorphism F and initiate a study of their semi-invariant submanifolds. These submanifolds are generalizations of CR submanifolds of almost complex geometry and semi-invariant submanifolds of several interesting geometries (almost product, almost contact and others). Characterization of the integrability of both invariant and anti-invariant distribution are done; the special case when F is covariant constant with respect to g.

1 Introduction

The geometry of manifolds endowed with geometrical structures has been intensively studied, and several important results have been published; see Yano-Kon [1]. The more important classes of such manifolds are formed by almost complex, almost product, almost contact, almost paracontact manifolds for which the cited book offers a good introduction. The geometry of submanifolds in these manifolds is very rich and interesting; as well, see for example the classical [2] or the more recent survey [3]. CR submanifolds introduced by Bejancu in [4] (for almost complex geometry) and [5] (for almost contact geometry), respectively, have had a great impact on the developing of the theory of submanifolds in these ambient manifolds; a proof of this fact is given by the books [6, 7].

In the present paper, we first introduce the concept of $(g, F, \mu )$-manifold which contains as particular cases all the above types of structures. Then, we study semi-invariant submanifolds of a $(g, F, \mu )$-manifold, which are extensions of CR submanifolds to this general class of manifolds. We find necessary and sufficient conditions for the integrability of both distributions on a semi-invariant submanifold; see Theorems 3.1 and 3.3. In particular, we prove that some semi-invariant submanifolds carry a natural foliation, Theorem 4.4 and we obtain characterizations of totally geodesic foliations on semi-invariant submanifolds in Theorems 4.8 and 4.10. For a particular value of the real parameter $\mu$ we can connect our study with the almost symplectic geometry and this fact opens some possible further applications in physical sciences having as example the relationship between CR-structures and Relativity pointed out in the last Chapter of [6].

2 Metric Geometry of Endomorphisms and Submanifolds

Let M be an m-dimensional manifold for which we denote by $C^{\infty }(M)$ the algebra of smooth functions on M and by $\Gamma (TM)$ the $C^{\infty }(M)$-module of smooth sections of the tangent bundle TM of M; let $X, Y, Z,\ldots$ denote such vector fields. We use the same notation $\Gamma (V)$ for any other vector bundle V over M. Let also ${\mathcal T}^1_1M$ be the $C^{\infty }(M)$-module of $\Gamma (TM\otimes T^*M)$ i.e. the real space of tensor fields of (1, 1)-type on M. Consider a fixed $F\in {\mathcal T}^1_1M$ usually called endomorphism or affinor or vector 1-form; the remarkable endomorphism of every manifold is the Kronecker tensor field $I=(\delta ^i_j)$.

Fix $\mu \in \{-1, +1\}$. Let now g be a Riemannian metric on M.

Definition 1.1

M is called a $(g, F, \mu )$-manifold if for any X, Y:

$$\begin{aligned} g(FX, Y)+\mu g(X, FY)=0. \end{aligned}$$

(1.1)

The geometry of the data $(M, g, F, \mu )$ is called endomorphism-metric geometry. If in particular, $F_{x}$ is nondegenerate at any point $x\in M$ then we say that M is a nondegenerate$(g, F, \mu )$-manifold; otherwise, M is called degenerate$(g, F, \mu )$-manifold.

Relation (1.1) says that the g-adjoint of F is $F^*=-\mu F$. In the literature, there is an abundance of examples of $(g, F, \mu )$-manifolds. Some of the main examples are presented here:

Example 1.2

• 1. An almost Hermitian manifold ([6, p. 11]) (M, g, J) is a nondegenerate $(g, F, \mu =+1)$-manifold; the nondegeneration is a consequence of $J^2=-I$.

• 2. An almost para-Hermitian manifold ([8]) (M, g, P) is a nondegenerate $(g, F, \mu =+1)$-manifold, while an almost Riemannian product manifold is a nondegenerate $(g, F, \mu =-1)$-manifold; the nondegeneration is a consequence of $P^2=I$.

• 3. An almost contact metric manifold ([6, p. 15]) $(M, g, \varphi , \xi , \eta )$ is a $(g, F, \mu =+1)$-manifold; as $\varphi (\xi )=0$, M is degenerate.

• 4. An almost paracontact manifold ([9]) $(M, g, \varphi , \xi , \eta )$ is a $(g, F, \mu =+1)$-manifold. As in the previous example, we have $\varphi (\xi )=0$ and therefore M is degenerate.

• 5. The general case of a nondegenerate $(g, F, \mu =+1)$-manifold is called structured manifold in [10].

Recall that a real 2k-dimensional manifold M is called an almost symplectic manifold if it is endowed with a nondegenerate 2-form $\Omega \in \Omega ^2(M)$. We derive the following characterization:

Proposition 1.3

LetMbe a$(g, F, \mu =+1)$-manifold. Then,Mis nondegenerate if and only if$\Omega$defined by:

$$\begin{aligned} \Omega (X, Y)=g(FX, Y) \end{aligned}$$

(1.2)

is an almost symplectic structure. In this case,mis even.

Proof

$\Omega$ is skew-symmetric from $\mu =+1$. A straightforward computation yields that $\Omega$ is nondegenerate if and only if M is a nondegenerate $(g, F, \mu =+1)$-manifold. $\square$

Example 1.4

For Example 1.2.1 $\Omega$ is exactly the fundamental or Kähler 2-form and then inspired by this fact we introduce:

Definition 1.5

For a nondegenerate $(g, F, \mu =+1)$-manifold $\Omega$ is call the fundamental 2-form.

In the last part of this section, let us recall briefly the geometry of Riemannian submanifolds. Consider an n-dimensional submanifold N of M. Then, the main objects induced by the Levi-Civita connection $\widetilde{\nabla }$ of (M, g) on N are involved in the well-known Gauss–Weingarten equations:

$$\begin{aligned} \widetilde{\nabla }_{X}Y=\nabla _{X}Y+h(X, Y), \quad \widetilde{\nabla }_{X}V=-A_{V}X+\nabla _{X}^{\perp }V, \end{aligned}$$

(1.3)

for any $X, Y\in \Gamma (TN)$ and $V\in \Gamma (T^{\bot }N)$. Here $\nabla$ is the Levi-Civita connection on N, h is the second fundamental form of N, $A_{V}$ is the Weingarten operator with respect to the normal section V and $\nabla ^{\bot }$ is the normal connection in the normal bundle $T^{\bot }N$ of N. Let us pointed out that h and $A_{V}$ are related by:

$$\begin{aligned} g(h(X, Y), V)=g(A_{V}X, Y). \end{aligned}$$

(1.4)

If h vanishes identically on N then N is called totally geodesic.

3 Submanifolds in Endomorphism-Metric Geometry

Next, we consider a submanifold N of a $(g, F, \mu )$-manifold M. Then, g induces a Riemannian metric on N which we denote by the same symbol g. Following the definition given by Bejancu [4] for CR submanifolds, we introduce a special class of submanifolds of M as follows:

Definition 2.1

N is a semi-invariant submanifold of M if there exists a distribution D on N satisfying the conditions:

• (i) D is F-invariant:

  $$\begin{aligned} F(D_{x})\subset D_{x}\ ,\ \ \ \forall \ x\in N. \end{aligned}$$

  (2.1)

• (ii) The complementary orthogonal distribution $D^{\perp }$ to D in TN is F-anti-invariant, that is:

  $$\begin{aligned} F(D_{x}^{\perp })\subset T_{x}^{\bot }N,\ \ \ \forall \ x\in N. \end{aligned}$$

  (2.2)

• (iii) $F^{2}(D^{\perp })$ is a distribution on N.

Some particular classes of semi-invariant submanifolds are defined as follows. Let p and q be the ranks of the distributions D and $D^{\perp }$, respectively. If $q=0$, that is $D^{\perp }=\left\{ 0\right\}$, we say that N is an F-invariant submanifold of M. If $p=0$, that is $D=\left\{ 0\right\}$, we call N an F-anti-invariant submanifold of M.

If $pq\ne 0$, then N is called a proper semi-invariant submanifold. Now, we denote by $\widetilde{D}$ the complementary orthogonal vector bundle to $F(D^{\perp })$ in $T^{\perp }N$. If $\widetilde{D}=\left\{ 0\right\}$ then we say that N is a normalF-semi-invariant submanifold.

Thus, N is an F-invariant, respectively F-anti-invariant, if and only if:

$$\begin{aligned} F(TN)\subset TN\ \ \ (\text {resp. }F(TN)\subset T^{\perp }N). \end{aligned}$$

(2.3)

N is normal F-semi-invariant if and only if:

$$\begin{aligned} F(D^{\perp })=T^{\perp }N. \end{aligned}$$

(2.4)

Examples 2.2

• 1. For Example 1.2.1, we obtain the classical concept of CR submanifold of Bejancu [6, p. 20]; the condition (iii) is satisfied from $J^2=-I$.

• 2. For Example 1.2.2, we obtain the notion of semi-invariant submanifold; for the almost para-Hermitian case; see [8], while for the second case; see [11]. The condition (iii) is satisfied again from $P^2=-I$.

• 3. For Example 1.2.3, we obtain the notion of semi-submanifold [6, p. 100] with $\xi \in T^{\bot }N$. This last condition implies $TN\subset {\mathrm{ker}} \ \eta$ and since $\varphi |_{{\mathrm{ker}} \ \eta }$ is an almost complex structure we get (iii). Semi-invariant $\xi ^{\bot }$-submanifolds of generalized quasi-Sasakian manifolds are studied in [12].

• 4. For Example 1.2.4, we obtain the concept of semi-submanifold from [13] with $\xi \in T^{\bot }N$. Again this condition means $TN\subset {\mathrm{ker}} \ \eta$ and since $\varphi |_{{\mathrm{ker}} \ \eta }$ is an almost product structure, we have (iii).

• 5. The condition (iii) does not appear in [10].

Returning to the Definition 2.1, we deduce that the tangent bundle TN and the normal bundle $T^{\bot }N$ of a semi-invariant submanifold N have the orthogonal decompositions:

$$\begin{aligned} TN=D\oplus D^{\perp }, \quad T^{\perp }N=F(D^{\perp })\oplus \widetilde{D}. \end{aligned}$$

(2.5)

Then, we denote by P and Q the projection morphisms of TN on D and $D^{\perp }$, respectively, and obtain for $X=PX+QX\in \Gamma (TN)$:

$$\begin{aligned} FX=\varphi X+\omega X \end{aligned}$$

(2.6)

where we put:

$$\begin{aligned} \varphi =F\circ P, \quad \omega =F\circ Q. \end{aligned}$$

(2.7)

Thus, $\varphi$ is a tensor field of (1, 1)-type on N, while $\omega$ is a $F(D^{\perp })-$valued vector 1-form on N. Hence we derive:

Proposition 2.3

LetNbe a semi-invariant submanifold of a$(g, F, \mu )$-manifoldM. Then:

• (iv) Nis a$(g, \varphi , \mu )$-manifold.

• (v) $F^{2}(D^{\perp })$is a vector subbundle of$D^{\perp }$.

• (vi) The vector bundle$\widetilde{D}$isF-invariant i.e. for all$x\in N$we have: $F(\widetilde{D}_{x})\subset \widetilde{D}_{x}$.

Proof

• (iv) By definition, g is a Riemannian metric on N and $\varphi$ is a tensor field of (1, 1)-type on N; we need only to show (2.1). By using (1.1) for F we obtain for $X, Y\in \Gamma (TN)$:

  $$\begin{aligned}&g(\varphi X, Y)=g(FPX, Y)=g(FPX, PY)\\&\quad =-\mu g(PX, FPY)=-\mu g(X, FPY)=-\mu g(X, \varphi Y). \end{aligned}$$

• (v) Take $X\in \Gamma (D)$ and $Y\in \Gamma (D^{\perp })$ in (2.1): $g(X, F^{2}Y)=-\mu g(FX, FY)=0$ since $FX\in \Gamma (D)$ and $FY\in \Gamma (T^{\perp }N)$. Hence, $F^{2}(D^{\perp })$ is orthogonal to D and by condition (iii), we deduce that $F^{2}(D^{\perp })$ is a vector subbundle of $D^{\perp }$.

• (vi) Take $X\in \Gamma (TN)$, $Y\in \Gamma (D^{\perp })$ and $V\in \Gamma (\widetilde{D})$. Then we obtain:

  $$\begin{aligned} g(FV, X)=-\mu g(V, FX)=-\mu g(V, \varphi X+\omega X)=0 \end{aligned}$$

  and:

  $$\begin{aligned} g(FV, FY)=-\mu g(V, F^{2}Y)=0 \end{aligned}$$

  since $\varphi X\in \Gamma (D)$, $\omega X\in \Gamma (FD^{\perp })$ and $F^{2}Y\in \Gamma (D^{\perp })$. Thus $F\widetilde{D}$ is orthogonal to $TN\oplus FD^{\perp }$, that is $F\widetilde{D}$ is a vector subbundle of $\widetilde{D}$. This completes the proof of the proposition.

$\square$

In the non-degenerated case, we have equalities for the above inclusions:

Corollary 2.4

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu )$-manifoldM. Then:

• (1) the above distributions satisfy:

  $$\begin{aligned} F(D)=D, \quad F^{2}(D^{\perp })=D^{\perp }, \quad F(\widetilde{D})=\widetilde{D}. \end{aligned}$$

  (2.8)

• (2) if$\mu =+1$then$D^{\bot }$and$F(D^{\bot })$are Lagrangian distribution on$(TM, \Omega )$. In particular ifNis a normal semi-invariant submanifold, then$T^{\bot }N$is a Lagrangian submanifold in$(TM, \Omega )$.

Proof

We need to prove only (2).

• (2.1) Let $X, Y\in \Gamma (D^{\bot })$; then $\Omega (X, Y)=g(FX, Y)=0$ since $FX\in \Gamma (T^{\bot }N)$ while $Y\in \Gamma (TN)$.

• (2.2) Let $X, Y\in \Gamma (F(D^{\bot }))$; then $\Omega (X, Y)=g(FX, Y)=0$ since $FX\in \Gamma (TN)$ while $Y\in \Gamma (T^{\bot }N)$.

$\square$

The second part of the above Corollary is extremely important since it relates the geometry of semi-invariant submanifolds with the almost symplectic geometry, a topic very studied from the point of view of applications in Analytical Mechanics.

4 Integrability of Distributions on a Semi-invariant Submanifold

Let N be a semi-invariant submanifold of a $(g, F, \mu )$-manifold M. Then, we recall that the Nijenhuis tensor field of F is defined as follows ([6, p.11]):

$$\begin{aligned} N_F(X, Y)=[FX, FY]+F^{2}[X, Y]-F[FX, Y]-F[X, FY], \end{aligned}$$

(3.1)

for any $X, Y\in \Gamma (TM)$. In a similar way, the Nijenhuis tensor field of $\varphi$ on N is given by:

$$\begin{aligned} N_{\varphi }(X, Y)=[\varphi X, \varphi Y]+\varphi ^{2}[X, Y]-\varphi [\varphi X, Y]-\varphi [X, \varphi Y], \end{aligned}$$

(3.2)

for any $X, Y\in \Gamma (TN)$. We recall that a tensor field of (1, 1)-type defines an integrable structure on a manifold if and only if its Nijenhuis tensor field vanishes identically on the manifold. Now we obtain necessary and sufficient conditions for the integrability of D and $D^{\perp }$ in terms of Nijenhuis tensor fields of F and $\varphi$.

Theorem 3.1

LetNbe a semi-invariant submanifold of a$(g, F, \mu )$-manifoldM. Then, the following assertions are equivalent:

• (1) Dis an integrable distribution.

• (2) The Nijenhuis tensor field of$\varphi$satifies:

  $$\begin{aligned} Q\circ N_{\varphi }=0, \quad \forall X, Y\in \Gamma (D). \end{aligned}$$

  (3.3)

• (3) The Nijenhuis tensor fields ofFand$\varphi$satisfy the equality: $N_F=N_{\varphi }$onD.

Proof

Firstly, we note that D is integrable if and only if:

$$\begin{aligned} Q([X, Y])=0,\quad \forall X, Y\in \Gamma (D). \end{aligned}$$

(3.4)

Since the last three terms in the right side of (3.2) lie in $\Gamma (D)$ we deduce that:

$$\begin{aligned} Q\circ N_{\varphi }(X, Y)=Q([FX, FY]),\quad \forall X, Y\in \Gamma (D). \end{aligned}$$

(3.5)

As M is nondegenerate we deduce that $\varphi$ is an automorphism on $\Gamma (D)$. Thus, the equivalence of (1) and (2) it follows directly. Next, we obtain for any $X, Y\in \Gamma (D)$:

$$\begin{aligned} N_F(X, Y)=N_{\varphi }(X, Y)+F\omega ([X, Y])-\omega ([\varphi X, Y])-\omega ([X, \varphi Y]). \end{aligned}$$

(3.6)

If D is integrable, then the last three terms of (3.6) vanishes and this yields 3). Conversely, suppose that $N_F=N_{\varphi }$ on D; then:

$$\begin{aligned} F\omega ([X, Y])=\omega ([\varphi X, Y]+[X, \varphi Y]). \end{aligned}$$

(3.7)

Obviously, the right-hand-side of the previous equation is in $\Gamma (F(D^{\bot }))\subset \Gamma (T^{\bot }N)$. On the other hand, the left-hand-side is in $\Gamma (F^2D^{\bot })\subset \Gamma (TN)$; we conclude that both sides in (3.7) must vanish. Finally, from: $F^{2}Q([X,Y])=0$ and $F^2$ automorphism of $\Gamma (TM)$, we deduce 1). $\square$

Remark 3.2

For Example 1.2.1, the equivalence of (1) and (2) is exactly the Theorem 2.2. of [6, p. 25], while the equivalence of (1) and (3) is the Theorem 2.1. of [6, p. 25].

Now, we consider $X, Y\in \Gamma (D^{\bot })$. Then, taking into account that $\varphi X=\varphi Y=0$, we get:

$$\begin{aligned} N_{\varphi }(X, Y)=F^{2}P[X, Y] \end{aligned}$$

(3.8)

and this enables us to state the following:

Theorem 3.3

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu )-$manifold. Then$D^{\bot }$is integrable if and only if the Nijenhuis tensor field of$\varphi$vanishes identically on$D^{\bot }$.

Remark 3.4

For Example 1.2.1, the above result is the Theorem 2.3. of [6, p. 26].

5 A Natural Foliation on a Semi-invariant Submanifold

Let $\widetilde{\nabla }$ be the Levi-Civita connection on M with respect to the Riemannian metric g. Then, F is a parallel tensor field on M if:

$$\begin{aligned} \widetilde{\nabla }F=0. \end{aligned}$$

(4.1)

Examples 4.1

• (1) For Example 1.2.1, we have the notion of Kähler manifold.

• (2) For Example 1.2.2, the first part we have the concept of para-Kähler manifold while for the second part the notion of locally Riemannian product manifold.

• (3) For Example 1.2.3, we get the notion of cosymplectic manifold.

In the present section, we study the geometry of semi-invariant submanifolds of $(g, F, \mu )$-manifolds with parallel tensor field F. First, we prove the following:

Proposition 4.2

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu )$-manifold with parallel tensor fieldF. Then, for all$X, Y\in \Gamma (D^{\bot })$:

$$\begin{aligned} A_{FX}Y-A_{FY}X=\varphi ([X, Y]). \end{aligned}$$

(4.2)

Proof

By using the Weingarten equation and the parallelism condition, we get:

$$\begin{aligned} A_{FX}Y=\nabla ^{\bot }_{Y}FX-\widetilde{\nabla }_{Y}FX=\nabla ^{\bot }_{Y}FX-F(\widetilde{\nabla }_XY). \end{aligned}$$

(4.3)

Writing a similar equation by interchanging X and Y and then subtracting, we obtain:

$$\begin{aligned} A_{FX}Y-A_{FY}X=\nabla _{Y}^{\bot }FX-\nabla _{X}^{\bot }FY+F([X, Y]), \end{aligned}$$

(4.4)

since $\nabla$ is a torsion-free linear connection. Thus, (4.2) is obtained by equalizing the tangent parts to N in the above equation. $\square$

Example 4.3

Relation (4.2) becomes, for Example 1.2.1, equation (2.2) of [6, p. 43].

Now, we can state the following main result:

Theorem 4.4

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu =+1)$-manifold with parallel tensor fieldF. Then, theF-anti-invariant distribution$D^{\bot }$is integrable.

Proof

For any $X, Y\in \Gamma (D^{\bot })$ and $Z\in \Gamma (D)$ we have:

$$\begin{aligned} g(A_{FX}Y, Z)& = -g(F\widetilde{\nabla }_{Y}X, Z)=+\mu g(\widetilde{\nabla }_{Y}X, FZ)=-\mu g(X, \widetilde{\nabla }_{Y}FZ)\nonumber \\ & = \mu ^2g(FX, \widetilde{\nabla }_{Y}Z)=\mu ^2g(FX, [Y,Z]+\widetilde{\nabla }_{Z}Y)\nonumber \\ & = \mu ^2g(FX, \widetilde{\nabla }_{Z}Y). \end{aligned}$$

(4.5)

Also, we have:

$$\begin{aligned}&g(A_{FY}X, Z)=\mu ^2g(FY, \widetilde{\nabla }_{Z}X)\nonumber \\&\quad =-\mu ^2g(F\widetilde{\nabla }_{Z}Y, X)=\mu ^3g(\widetilde{\nabla }_{Z}Y, FX). \end{aligned}$$

(4.6)

Comparing (4.5) and (4.6), we deduce that for $\mu =+1$:

$$\begin{aligned} g(A_{FX}Y-A_{FY}X, Z)=0 \end{aligned}$$

which means that $A_{FX}Y-A_{FY}X\in \Gamma (D^{\bot })$. On the other hand, from (4.2) we conclude that:

$$\begin{aligned} A_{FX}Y-A_{FY}X\in \Gamma (D). \end{aligned}$$

and thus we have that:

$$\begin{aligned} A_{FX}Y-A_{FY}X=0. \end{aligned}$$

(4.7)

Finally, returning to (4.2) and taking into account that F is nondegenerate we deduce that:

$$\begin{aligned} P[X, Y]=0, \end{aligned}$$

that is, $D^{\bot }$ is integrable. $\square$

Remark 4.5

For Example 1.2.1, the above result is part (i) of Theorem 1.1. of [6, p. 39].

Regarding the integrability of D, we prove the following:

Theorem 4.6

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu )$-manifoldMwith parallel tensor fieldF. Then, theF-invariant distributionDis integrable if and only if the second fundamental formhofNsatisfies for any$X, Y\in \Gamma (D)$and$Z\in \Gamma (D^{\bot })$:

$$\begin{aligned} g(h(X, \varphi Y)-h(Y, \varphi X), FZ)=0. \end{aligned}$$

(4.8)

Proof

By using the Gauss equation we deduce that:

$$\begin{aligned} \nabla _{X}\varphi Y+h(X, \varphi Y)=\varphi (\nabla _{X}Y)+\omega (\nabla _{X}Y)+Fh(X,Y). \end{aligned}$$

(4.9)

Write a similar equation by interchanging X and Y, and then subtracting we obtain:

$$\begin{aligned} \nabla _{X}\varphi Y-\nabla _{Y}\varphi X+h(X, \varphi Y)-h(Y, \varphi X)=\varphi ([X, Y])+\omega ([X,Y]) \end{aligned}$$

(4.10)

since h is symmetric and $\nabla$ is a torsion-free linear connection. Equalize the normal parts in the above equation and obtain:

$$\begin{aligned} h(X, \varphi Y)-h(Y, \varphi X)=\omega ([X, Y]). \end{aligned}$$

(4.11)

Now, suppose that D is integrable; then (4.8) is immediately. Conversely, if (4.8) is satisfied, then with (4.11) we deduce that:

$$\begin{aligned} 0=-\mu g(Q[X, Y], F^{2}Z). \end{aligned}$$

Since F is nondegenerate we infer that $F^{2}$ is an automorphism of $\Gamma (D^{\bot })$ and hence D is integrable. $\square$

Remark 4.7

In particular, if F is an almost complex structure on M, then we obtain the results of Bejancu [6] and Blair–Chen [14], respectively, for CR submanifolds.

Now, for $\mu =+1$ we denote by $\mathcal {F}^{\bot }$ the natural foliation defined by the F-anti-invariant distribution $D^{\bot }$ and call it the F-anti-invariant foliation on N. We recall that $\mathcal {F}^{\bot }$ is called a totally geodesic foliation if each leaf of $\mathcal {F}^{\bot }$ is totally geodesic immersed in N. Thus, $\mathcal {F}^{\bot }$ is totally geodesic if and only if the Levi-Civita connection $\nabla$ of N satisfies for all $Y, Z\in \Gamma (D^{\bot })$:

$$\begin{aligned} \nabla _{Y}Z\in \Gamma (D^{\bot }). \end{aligned}$$

(4.12)

Theorem 4.8

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu )$-manifoldMwith parallel tensor fieldF. Then the following assertions are equivalent:

• (i) TheF-anti-invariant foliation is totally geodesic.

• (ii) The second fundamental formhofNsatisfies for all$X\in \Gamma (D)$and$Y\in \Gamma (D^{\perp })$:

  $$\begin{aligned} h(X, Y)\in \Gamma (\widetilde{D}). \end{aligned}$$

  (4.13)

• (iii) $D^{\bot }$is$A_{V}$-invariant for any$V\in \Gamma (FD^{\bot })$that is we have for all$Y\in \Gamma (D^{\bot })$:

  $$\begin{aligned} A_{V}Y\in \Gamma (D^{\bot }). \end{aligned}$$

  (4.14)

Proof

We have for any $X\in \Gamma (D)$ and $Y, Z\in \Gamma (D^{\bot })$:

$$\begin{aligned}&g(\nabla _{Y}Z, FX)=g(\widetilde{\nabla }_{Y}Z, FX)\nonumber \\&\quad =-\mu g(\widetilde{\nabla }_{Y}FZ, X)=\mu g(A_{FZ}Y, X)=\mu g(h(X, Y), FZ). \end{aligned}$$

(4.15)

Now, suppose that $\mathcal {F}^{\bot }$ is totally geodesic; then the first term of (4.15) vanishes. Hence, the last term in (4.15) vanishes which implies ii). Conversely, suppose (4.14) is satisfied. Then from (4.15), we deduce (4.12) since F is an automorphism of $\Gamma (D)$. This proves the equivalence of (i) and (ii). The equivalence of (ii) and (iii) is straightforward. $\square$

Remark 4.9

For Example 1.2.1, the equivalence of (i) and (ii) is the Theorem 1.3. of [6, p. 41].

Finally, we can prove the following:

Theorem 4.10

LetNbe a semi-invariant submanifold of a nondegenerate$(g, F, \mu )$-manifold with parallel tensor fieldF. Then, theF-invariant distributionDis integrable and the foliation$\mathcal {F}$defined byDis totally geodesic if and only if the second fundamental formhofNsatisfies for all$X, Y\in \Gamma (D)$:

$$\begin{aligned} h(X, Y)\in \Gamma (\widetilde{D}). \end{aligned}$$

(4.16)

Proof

D is integrable and $\mathcal {F}$ is totally geodesic if and only if for all $X, U\in \Gamma (D)$:

$$\begin{aligned} \nabla _{X}U\in \Gamma (D). \end{aligned}$$

(4.17)

This is equivalent to:

$$\begin{aligned} g(\widetilde{\nabla }_{X}U, Z)=0, \end{aligned}$$

(4.18)

for all $Z\in \Gamma (D^{\bot })$. As F is an automorphism of $\Gamma (D)$, we can write the above equality as follows:

$$\begin{aligned} g(\widetilde{\nabla }_{X}FY, Z)=0, \end{aligned}$$

(4.19)

for all $X, Y\in \Gamma (D)$ and $Z\in \Gamma (D^{\bot })$, which is equivalent to:

$$\begin{aligned} g(\widetilde{\nabla }_{X}Y,FZ)=0. \end{aligned}$$

(4.20)

By using the Gauss equation, the last relation is equivalent with:

$$\begin{aligned} g(h(X, Y), FZ)=0, \end{aligned}$$

(4.21)

which completes the proof of the theorem. $\square$

Remark 4.11

For Example 1.2.1, the above result is Theorem 1.2. of [6, p. 40].