Semi-Invariant Submanifolds of Riemannian Product Manifold

B. Şahin and M. Atçeken

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

In this paper, the geometry of submanifolds of a Riemannian product manifold is studied. Fundamental properties of these submanifolds are investigated such as integrability of distributions, totally umbilical semi-invariant submanifold. Finally, necessary and sufficient conditions are given on a semi-invariant submanifold of a Riemannian product manifold to be a locally Riemannian manifold.

Mathematics Subject Classification: 53C42, 53C15. Key words: Riemannian product manifold, totally umbilical submanifold, mixed geodesic submanifold, constant sectional curved manifold

1 Introduction

The geometry of a submanifold (M, g) of a locally Riemannian product manifold (\overline{M}_{1} × \overline{M}_{2}, \overline{g}_{1} ⊗ \overline{g}_{2}) was widely studied by many geometers. In particularly, K. Matsumoto has proved that (M, g) is a locally product Riemannian manifold of Riemannian manifolds (M_{a}, g_{a}) and (M_{b}, g_{b}), if it is an invariant submanifold of a Riemannian product manifold (\overline{M}_{1} × \overline{M}_{2}, \overline{g}_{1} ⊗ \overline{g}_{2})(see [4]). After then Senlin, Xu., and Yilong, Ni., have updated theorem of Matsumoto and proved that M_{a} ⊂ \overline{M}_{1} and M_{b} ⊂ \overline{M}_{2}. Moreover, they have proved that (M_{a}, g_{a}) and (M_{b}, g_{b}) are pseudoumbilical submanifolds of (\overline{M}_{1}, \overline{g}_{1}) and (\overline{M}_{2}, \overline{g}_{2}), respectively, if (M, g) is a pseudoumbilical submanifold of (\overline{M}, \overline{g}) = (\overline{M}_{1} × \overline{M}_{2}, \overline{g}_{1} ⊗ \overline{g}_{2}). They have also demonstrated that M is isometric to the production of its two totally geodesic submanifolds (M_{a}, g_{a}) and (M_{b}, g_{b}) which are submanifolds of (\overline{M}_{1}, \overline{g}_{1}) and (\overline{M}_{2}, \overline{g}_{2}), respectively (see [5]).

In this work, we study the geometry of semi-invariant submanifolds of a Riemannian manifold and proved that a semi-invariant submanifold of a Riemannian product manifold is a locally Riemannian product manifold iff A_{F D^{⊥}}D = 0, which is equivalent to ∇f = 0, or Bh(X, Y) = 0 for any X ∈ Γ(T M) and Y ∈ Γ(D). Moreover, necessary and sufficient conditions are given on distributions D and D^{⊥} of a semi-invariant submanifold M are integrable. Finally, we show that there exists no totally umbilical semi-invariant submanifold of positively or negatively curved Riemannian product manifold. Also we give an example for semi-invariant submanifold to illustrate the our results.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 91-100.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

2 Preliminaries

In this section, we give some notations and terminology used througthout this paper. We recall some necessary facts and formulas from the theory of submanifolds. For an arbitrary submanifold M of a Riemannian manifold \overline{M}, Gauss and Weingarten formulas are given by

\overline{∇}_{X}Y = ∇_{X}Y + h(X, Y), (2.1)

and

\overline{∇}_{X}ξ = −A_{ξ}X + ∇_{X}^{⊥}ξ, (2.2)

respectively, where \overline{∇}, ∇ are Levi-Civita connections on the Riemannian manifolds \overline{M} and its submanifold M, respectively, and X, Y are vector fields tangent to M, ξ is a vector field normal to M, h : T M × T M ⟶ T M^{⊥} is the second fundamental form of M, ∇^{⊥} is the normal connection in the normal vector bundle T M^{⊥}, and A_{ξ} is the shape operator of the second quadratic form for a normal vector ξ. Moreover, we have

g(A_{ξ}X, Y) = \overline{g}(h(X, Y), ξ), (2.3)

where the symbols \overline{g} and g mean the Riemannian metrics of \overline{M} and its submanifold M, respectively.

We denote the Riemannian curvature tensors of the Levi-Civita connections \overline{∇} and ∇ on \overline{M} and M by \overline{R} and R, respectively. The Gauss, Codazzi, and Ricci equations are given by

\overline{g}(\overline{R}(X, Y)Z, W) = \overline{g}(R(X, Y)Z, W) + \overline{g}(h(X, W), h(Y, Z))
(2.4)  − \overline{g}(h(X, Z), h(Y, W))

(2.5)  (\overline{R}(X, Y)Z)^{⊥} = (\overline{∇}_{X}h)(Y, Z) − (\overline{∇}_{Y} h)(X, Z),

(2.6)  \overline{g}(\overline{R}(X, Y)ξ, η) = \overline{g}(\overline{R}^{⊥}(X, Y)ξ, η) − \overline{g}([A_{ξ}, A_{η}]X, Y)

respectively, where the vector fields X, Y, Z, W are tangent to M, the vector fields ξ and η are orthogonal to M, (\overline{R}(X, Y)Z)^{⊥} denotes the normal of \overline{R}(X, Y)Z and the derivative \overline{∇}h is defined by

(\overline{∇}_{X}h)(Y, Z) = (∇_{X}^{⊥}h)(Y, Z) − h(∇_{X}Y, Z) − h(∇_{X}Z, Y). (2.7)

Semi-Invariant Submanifolds of Riemannian Product Manifold 93 We recall that M is called a curvature-invariant submanifold, if it has (\overline{R}(X, Y)Z)^{⊥} = 0, (2.8) which is equivalent to

(\overline{∇}_{X}h)(Y, Z) = (\overline{∇}_{Y} h)(X, Z),

for all X, Y, Z ∈ Γ(T M) [3]. Definition 2.1 For a submanifold M ⊆ \overline{M} the mean-curvature vector field H is defined by the formula

H = \frac{1}{n} ∑_{i=1}^{n} h(e_{i}, e_{i}), (2.9)

where \{e_{i}\} is a local orthonormal basis in T M. If a submanifold M ⊆ \overline{M} having one of the conditions

h = 0, h(X, Y) = g(X, Y)H, g(h(X, Y), H) = λg(X, Y),

H = 0, λ ∈ C^{∞}(M, R), (2.10) then it is called totally geodesic, totally umbilical, pseudo-umbilical and minimal, respectively [2].

Let (\overline{M}_{1}, \overline{g}_{1}) and (\overline{M}_{2}, \overline{g}_{2}) be Riemannian manifolds with dimensions n_{1} and n_{2}, respectively. Then \overline{M} = \overline{M}_{1} ×\overline{M}_{2} is the Riemannian product manifold of Riemannian manifolds \overline{M}_{1} and \overline{M}_{2}. We denote the projection mappings of T (\overline{M}_{1} × \overline{M}_{2}) to T \overline{M}_{1} and T \overline{M}_{2} by π_{∗} and σ_{∗}, respectively. Then we have π_{∗} + σ_{∗} = I, π_{∗}^{2} = π_{∗}, σ_{∗}^{2} = σ_{∗}, π_{∗} × σ_{∗} = σ_{∗} × π_{∗} = 0. (2.11) Then the Riemannian metric of \overline{M}_{1} × \overline{M}_{2} is given by \overline{g}(X, Y) = \overline{g}_{1}(π_{∗}X, π_{∗}Y) + \overline{g}_{2}(σ_{∗}X, σ_{∗}Y) (2.12) for all X, Y ∈ Γ(T (\overline{M}_{1} × \overline{M}_{2})). Set F = π_{∗} − σ_{∗}, then we can easily see that F^{2} = I. It follows \overline{g}(X, Y) = \overline{g}(F X, F Y) (2.13) for all X, Y ∈ Γ(T (\overline{M}_{1} × \overline{M}_{2})).

By the definition of \overline{g}, \overline{M}_{1} and \overline{M}_{2} are totally geodesic submanifolds of \overline{M}_{1} × \overline{M}_{2}. We denote the Levi-Civita connection of \overline{M}_{1} × \overline{M}_{2} by \overline{∇}, we can easily see that (\overline{∇}_{X}F)Y = 0. (2.14) for any X, Y ∈ Γ(T (\overline{M}_{1} × \overline{M}_{2}))(For the detail, we refer to[5]).

3 Semi-invariant submanifold of a Riemannian product manifold

We denote the Riemannian product manifold (\overline{M}_{1} × \overline{M}_{2}, \overline{g}_{1} × \overline{g}_{2}) by (\overline{M}, \overline{g}) througthout this paper.

Definition 3.1 let M be a submanifold of a Riemannian product manifold \overline{M}. We suppose that M has two the distributions such as D and D^{⊥} such that T M = D⊕D^{⊥}, F (D) = D and F (D^{⊥}) ⊂ T M^{⊥}. In this case, M is called semi-invariant submanifold of \overline{M}.

In the rest of this paper, we assume that M semi-invariant submanifold of \overline{M}. We denote the orthogonal complementary of F (D^{⊥}) in T M^{⊥} by V, then we have direct sum

T M^{⊥} = F (D^{⊥}) ⊕ V.

We denote the projection mappings of T M to D and D^{⊥} by P and Q, respectively. Then for each X tangent to T M, we can write F X in the following way:

F X = fX + ωX, (3.1)

where fX = F P X and ωX = F QX are respectively the tangent part and the normal part of F X. Also, for each vector field ξ normal to M, we put

F ξ = Bξ + Cξ, (3.2)

where Bξ and Cξ are respectively the tangent part and the normal part of F ξ.

We denote dimensions of the distributions D and D^{⊥} by p and q, respectively. Then for q = 0(resp. p = 0) a semi-invariant submanifold becomes an invariant submanifold(resp. an anti-invariant submanifold). A proper semi-invariant submanifold is a semi-invariant submanifold which is neither an invariant submanifold nor an anti-invariant submanifold.

Example 3.2 We consider a submanifold M in R^{6} given by the equations:

X_{1} = X_{6} + \frac{1}{2}(X_{3} + X_{4})^{2}, X_{2} = X_{5}.

It is easy check that M is a semi-invariant submanifold of R^{6} = R^{3} × R^{3}. Then by direct calculation we obtain

T M = Span\{U_{1} = \frac{∂}{∂X_{2}} + \frac{∂}{∂X_{5}} , U_{2} = (X_{3} + X_{4})\frac{∂}{∂X_{1}} + \frac{∂}{∂X_{3}} ,

U_{3} = (X_{3} + X_{4})\frac{∂}{∂X_{1}} + \frac{∂}{∂X_{4}} , U_{4} = \frac{∂}{∂X_{1}} + \frac{∂}{∂X_{6}} \}

and

T M^{⊥} = Span\{ξ_{1} = −\frac{∂}{∂X_{1}} + (X_{3} + X_{4})\frac{∂}{∂X_{3}} + (X_{3} + X_{4})\frac{∂}{∂X_{4}} + \frac{∂}{∂X_{6}} ,

ξ_{2} = \frac{∂}{∂X_{2}} − \frac{∂}{∂X_{5}} \},

where D = Span{U_{2}, U_{3}, U_{4}} and D^{⊥} = Span{U_{1}}.

Semi-Invariant Submanifolds of Riemannian Product Manifold 95 Definition 3.3 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then M is called mixed-geodesic semi-invariant submanifold if h(X, Y) = 0 for any X ∈ Γ(D) and Y ∈ Γ(D^{⊥}). We denote the Levi-Civita connections on M and \overline{M} by ∇ and \overline{∇}, respectively. Proposition 3.4 Let \overline{M} be a Riemannian product manifold and M be a semiinvariant submanifold of \overline{M}. Then we have A_{F Z}W = −A_{F W} Z, (3.3) for all Z, W ∈ Γ(D^{⊥}) Proof. From (2.1), (2.2), (2.14) and (3.1) we have

−A_{F Z}X + ∇_{X}^{⊥}F Z = F ∇_{X}Z + F h(X, Z)

for any X ∈ Γ(T M) and Z ∈ Γ(D^{⊥}). Using (2.13) we obtain

−\overline{g}(A_{F Z}X, W) = \overline{g}(h(X, Z), F W),

for any W ∈ Γ(D^{⊥}). Since A is self adjoint, from (2.3) we get

−\overline{g}(A_{F Z}W, X) = \overline{g}(A_{F W} Z, X),

which proves our assertion. Lemma 3.5 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then we have A_{ξ}F X = A_{F ξ}X (3.4) for any X ∈ Γ(D) and ξ ∈ Γ(V). Proof. Since \overline{∇} is the Levi-Civita connection, from (2.14) we derive

\overline{g}(h(F X, Y), ξ) = −\overline{g}(\overline{∇}_{Y} F ξ, X),

for any X ∈ Γ(D), Y ∈ Γ(T M) and ξ ∈ Γ(V). Using (2.2) and (2.3) we get

\overline{g}(A_{ξ}F X, Y) = \overline{g}(A_{F ξ}X, Y).

Thus proof is complete. Lemma 3.6 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then we have ∇_{Z}^{⊥}F W − ∇_{W}^{⊥} F Z ∈ Γ(D^{⊥}), (3.5) for any Z, W ∈ Γ(D^{⊥}).

96 B.Şahin and M.Atçeken Proof. From (2.1) and (2.2) we have \overline{g}(A_{F ξ}Z, W) = \overline{g}(∇_{Z}^{⊥}F W, ξ) (3.6) for any W, Z ∈ Γ(D^{⊥}) and ξ ∈ Γ(V). Since A is self adjoint, from (3.6) we get

\overline{g}(∇_{Z}^{⊥}F W − ∇_{W}^{⊥} F Z, ξ) = 0,

which gives (3.5). Theorem 3.7 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then D^{⊥} is integrable if and only if h(X, W) ∈ Γ(V) (3.7) for any X ∈ Γ(D) and W ∈ ΓD^{⊥}. Proof. From (2.2), (2.14) and (3.3) we get

F [Z, W] = 2A_{F Z}W + ∇_{Z}^{⊥}F W − ∇_{W}^{⊥} F Z

for any Z ∈ Γ(D^{⊥}). Thus from (2.3) and (2.13) we derive

\overline{g}([Z, W], F X) = 2\overline{g}(h(W, X), F Z).

Hence the proof is complete. Theorem 3.8 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then D is integrable if and only if h(X, F Y) = h(Y, F X) (3.8) for any X, Y ∈ Γ(D). Proof. By using (2.1), (2.2), (2.14) and (3.1) we derive

∇_{X}F Y + h(X, F Y) = P ∇_{X}Y + ω∇_{X}Y + F h(X, Y),

where interchanging role of vector fields X and Y, we obtain

∇_{Y} F X + h(Y, F X) = P ∇_{Y} X + ω∇_{Y} X + F h(Y, X).

Thus we have

h(X, F Y) − h(F X, Y) = ω([X, Y]).

This completes the proof of the theorem. Lemma 3.9 Let \overline{M} be a Riemannian product manifold and M be a mixed-geodesic semi-invariant submanifold of \overline{M}. Then we have A_{F ξ}X = F A_{ξ}X (3.9) for any X ∈ Γ(D) and ξ ∈ Γ(V).

Proof. From (2.1) and (2.2) we have

\overline{g}(A_{F ξ}X − F A_{ξ}X, Y) = \overline{g}(A_{F ξ}X, Y) − \overline{g}(A_{ξ}X, F Y)

for any X ∈ Γ(D), Y ∈ Γ(D^{⊥}) and ξ ∈ Γ(V). Since M is a mixed-geodesic submanifold, we have A_{F ξ}X ∈ Γ(D). Thus using the equation (2.3) we obtain

\overline{g}(A_{F ξ}X − F A_{ξ}, Y) = 0.

On the other hand, from (2.3) we get

\overline{g}(A_{F ξ}X − F A_{ξ}X, Z) = \overline{g}(h(X, Z), F ξ) − \overline{g}(h(X, F Z), ξ)

for any X, Z ∈ Γ(D). Thus from (2.14) we derive

\overline{g}(A_{F ξ}X − F A_{ξ}, Z) = 0,

which proves our assertion.

Theorem 3.10 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then M is a locally Riemannian product manifold if and only if A_{F Z}X = 0 for all X ∈ Γ(D) and Z ∈ Γ(D^{⊥}).

Proof. Let M be a semi-invariant submanifold of a Riemannian product manifold (\overline{M}, \overline{g}). Then from (2.1) and (2.2) we have

\overline{g}(∇_{X}F Y, Z) = \overline{g}(A_{F Z}X, Y) (3.10)

and

\overline{g}(∇_{W} Z, F X) = −\overline{g}(A_{F Z}X, W) (3.11)

for any X, Y ∈ Γ(D) and Z, W ∈ Γ(D^{⊥}). Now, we suppose that M is a locally Riemannian product manifold. Then the distributions D and D^{⊥} are parallel. From (3.10) and (3.11) we have A_{F Z}X ∈ Γ(D) and A_{F Z}X ∈ Γ(D^{⊥}). Since D ∩ D^{⊥} = {0} we obtain A_{F Z}X = 0.

Conversely, if A_{F Z}X = 0, then from (3.10) and (3.11) we have the distributions D and D^{⊥} are integrable and leaves of them are parallel. This completes the proof of the theorem.

Proposition 3.11 Any pseudo umbilical proper semi-invariant submanifold of a Riemannian product manifold is a mixed-geodesic submanifold.

Proof. We suppose that M is a pseudo-umbilical proper semi-invariant submanifold of a Riemannian product manifold (\overline{M}, \overline{g}). Then we have

\overline{g}(h(X, Z), H) = \overline{g}(H, H)g(X, Z) = 0,

for all X ∈ Γ(D) and Z ∈ Γ(D^{⊥}), which implies that h(X, Z) = 0.

Theorem 3.12 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M}. Then M is a locally Riemannian product manifold if and only if ∇f = 0.

Proof. Let M be a locally Riemannian product semi-invariant submanifold of \overline{M}. Then we have ∇_{U}Y ∈ Γ(D) for all U ∈ Γ(T M) and Y ∈ Γ(D). Thus from (2.2) and (3.1) we obtain

h(U, F Y) = F ∇_{U}Y + Bh(U, F Y) + Ch(U, F Y) − ∇_{U}F Y

for any U ∈ Γ(T M) and Y ∈ Γ(D). Hence we get

h(U, F Y) = Ch(U, F Y)
(∇_{U}f)Y = 0
(3.12)  Bh(U, F Y) = 0.

In the similar way, we obtain (∇_{U}f)Z = 0 for any Z ∈ Γ(D^{⊥}).

Conversely, we suppose that ∇f = 0. Then we have ∇_{X}fY = f∇_{X}Y, for any X, Y ∈ Γ(D). It follows that ∇_{X}Y ∈ Γ(D). In the similar way ∇_{Z}W ∈ Γ(D^{⊥}) for any Z, W ∈ Γ(D^{⊥}). Thus M is a locally Riemannian product manifold.

Theorem 3.13 Let M be a semi-invariant submanifold of a Riemannian product manifold \overline{M}. Then M is a locally Riemannian product manifold if and only if Bh(X, Y) = 0 for all X ∈ Γ(T M) and Y ∈ Γ(D).

Proof. We assume that M is a locally Riemannian product manifold. Then from (3.12) we have Bh(X, Y) = 0 for any X ∈ Γ(T M) and Y ∈ Γ(D).

Conversely, we assume that Bh(X, Y) = 0 for any X ∈ Γ(T M) and Y ∈ Γ(D). Then from (2.1), (2.14), (3.1) and (3.2) we get

∇_{X}fY + h(X, F Y) = f∇_{X}Y + ω∇_{X}Y + Bh(X, Y) + Ch(X, Y)

for any X ∈ Γ(T M) and Y ∈ Γ(D). Thus we derive (∇_{X}f)Y = 0, that is, ∇_{X}Y ∈ Γ(D). On the other hand, making use of (2.1), (2.2), (2.14), (3.1) and (3.2) we obtain

−A_{F Z}X + ∇_{X}^{⊥}F Z = f∇_{X}Z + ω∇_{X}Z + Bh(X, Z) + Ch(X, Z)

for any X ∈ Γ(T M) and Z ∈ Γ(D^{⊥}). Thus we obtain

−A_{F Z}X = f∇_{X}Z (3.13)

for any X ∈ Γ(T M) and Z ∈ Γ(D^{⊥}). By the using (2.3), (2.13) and (3.2) we derive

\overline{g}(f∇_{X}Z, Y) = −\overline{g}(Ch(X, Y), Z) = 0,

for all X ∈ Γ(T M), Y ∈ Γ(D) and Z ∈ Γ(D^{⊥}). Hence we have ∇_{X}Z ∈ Γ(D^{⊥}). Thus proof is complete.

In case F (D^{⊥}) = T M^{⊥}, we can give the following theorem.

Theorem 3.14 Let \overline{M} be a Riemannian product manifold and M be a semi-invariant submanifold of \overline{M} such that F (D^{⊥}) = T M^{⊥}. Then M is a locally Riemannian product manifold if and only if h(X, Y) = 0 for all X ∈ Γ(T M) and Y ∈ Γ(D).

Theorem 3.15 Let M be a totally umbilical proper semi-invariant submanifold of a Riemannian product manifold \overline{M}. Then one only of the following assertions are valid:

1) dimD^{⊥} = 1

2) M is a totally geodesic submanifold.

Proof. We suppose that M is a totally umbilical submanifold of a Riemannian product manifold \overline{M}. Then from (2.1) and (2.2) and (3.1) we have

−\overline{g}(A_{F W} Z, Z) = \overline{g}(F h(Z, W), Z)

for all Z, W ∈ Γ(D^{⊥}). Since M is a totally umbilical submanifold, from (2.3) and (2.13) we obtain

−\overline{g}(Z, Z)\overline{g}(F H, W) = \overline{g}(Z, Z)\overline{g}(F H, Z), (3.14)

where H is the mean curvature vector field of M in \overline{M}. Interchanging role of Z and W in (3.14) we get

−\overline{g}(W, W)\overline{g}(F H, Z) = \overline{g}(W, W)\overline{g}(F H, W). (3.15)

Thus from (3.14) and (3.15) we obtain

\overline{g}(F H, Z) = \frac{\overline{g}(Z, W)^{2}}{∥Z∥^{2}∥W ∥^{2}} \overline{g}(F H, Z). (3.16)

Hence, either \overline{g}(F H, Z) = 0 or Z and W are linearly dependent. If Z and W are linearly dependent, then dimD^{⊥} = 1.

We suppose that dimD^{⊥} > 1. Then from (3.3) we have

A_{F BH}Z = −A_{F Z}BH

for any Z ∈ Γ(D^{⊥}). By the using (2.3) we get

−\overline{g}(Z, W)\overline{g}(BH, BH) = \overline{g}(BH, W)\overline{g}(H, F Z).

Since dimD^{⊥} > 1, we can choose W orthogonal to BH. Then BH = 0, that is, H ∈ Γ(V). Now we assume that H ≠ 0. From (2.3) and (3.4) we derive

\overline{g}(F H, H)\overline{g}(X, Y) = \overline{g}(H, H)\overline{g}(F X, Y) (3.17)

for any X ∈ Γ(D). We note that the leaf of D is an invariant submanifold of Riemannian product manifold \overline{M}. We denote the leaf of D by N. Since N is an invariant submanifold of \overline{M}, it is a product manifold. Set N = N_{1} × N_{2}. Then we have

T N_{1} = {X ∈ Γ(T N)|F X = X}

and

T N_{2} = {X ∈ Γ(T N)|F X = −X}.

From (3.17) we obtain

\overline{g}(X, F X) = \overline{g}(X, X)

for any X ∈ Γ(D). Thus for X = X_{2} ∈ Γ(T N_{2}), we have

−\overline{g}(X_{2}, X_{2}) = \overline{g}(X_{2}, X_{2}),

i.e.,

∥X_{2}∥ = 0 ⟹ X_{2} = 0.

This is a contradiction.

Theorem 3.16 There exists no any totally umbilical proper semi-invariant submanifold of positively or negatively curved Riemannian product manifold \overline{M}.

Proof. We assume that Riemannian product manifold \overline{M} has constant sectional curvature c ≠ 0 and let M be a totally umbilical proper semi-invariant submanifold of \overline{M}. Then form the equations Gauss and Codazzi, we have

\overline{K}(X, Y, X, Y) = \overline{K}(X, Y, F X, F Y) = −\overline{g}(X, F X)\overline{g}(∇_{Y}^{⊥}H, F Y)
\overline{K}(X ∧ Y) = −\overline{g}(X, F X)\overline{g}(∇_{Y}^{⊥}H, F Y).

Since the vector fields X and F X are linearly independent, we can choose X orthogonal to F X. In this case, we obtain

\overline{K}(X ∧ Y) = 0.

This is a contradiction, where \overline{K} denotes the Riemannian-Christoffel curvature tensor of \overline{M}.

References

[1] M. Atc.eken and S. Keleş., On Product Riemannian Manifolds, Differential Geometry-Dynamical Systems, Vol.5 No.1, 2003, pp.1-7.

[2] B.Y. Chen, Geometry of Submanifolds, Marcel Dekker Inc., New York, 1973.

[3] B.Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publishing Co Pte Ltd., 1984.

[4] K. Matsumoto, On Submanifolds of Locally Product Riemannian Manifolds, TRU Mathematics 18-2, 1982, 145-157.

[5] X. Senlin and N. Yilong, Submanifolds of Product Riemannian Manifold, Acta Mathematica Scientia 2000, 20(B) 213-218.

[6] B. O’Neill, Semi-Riemannian Geometry, Academic Press Inc., 1983.

Inonu University, Faculty of Science and Art,

Department of Mathematics, 44100 Malatya/TURKEY email: matceken@inonu.edu.tr; bsahin@inonu.edu.tr