ON SUBMANIFOLDS OF A RIEMANNIAN MANIFOLD WITH A SEMI-SYMMETRIC RECURRENT-METRIC CONNECTION

We study some properties of submanifolds of a Riemannian manifold with a semi-symmetric recurrent-metric connection. Among others, the Gauss equation, the Codazzi-Mainardi equation and the Ricci equation for such a connection have been derived. AMS Subject Classification: 53A30, 53B35, 53C25, 53C55, 53C56


Introduction
Let M = (M, g) be a Riemannian manifold of dimension n with a metric tensor g.A linear connection ∇ on M satisfies where f, g are smooth functions on M and X, Y, Z are smooth vector fields on M .The torsion tensor T of ∇ is given by If the torsion tensor T vanishes, then ∇ is called symmetric, otherwise it is nonsymmetric.If the metric tensor g of M satisfies ∇g = 0, then ∇ is called metric, otherwise it is nonmetric.It is well known that the Levi-Civita connection is the only linear connection which is both symmetric and metric.In particular, a nonsymmetric connection is said to be semi-symmetric if its torsion tensor T is of the form where u is a 1-form on M .The idea of semi-symmetric metric connection on a Riemannian manifold was introduced by Yano and some of its properties were studied in [14].A hypersurface of a Riemannian manifold with a semi-symmetric metric connection was studied by Imai [6].Later in [9] Nakao investigated submanifolds of a Riemannian manifold with the semi-symmetric metric connection.On the other hand, Agashe and Chafle [1] introduced the idea of a semi-symmetric nonmetric connection on a Riemannian manifold and they [2] sequently studied submanifolds of a Riemannian manifold with the semi-symmetric nonmetric connection mentioned in [1].Other types of semi-symmetric nonmetric connections were introduced by Sengupta, De and Binh [12]; Sengupta and De [11].Later submanifolds of Riemannian manifolds with the semi-symmetric nonmetric connections defined in [11,12] were studied by Ozgur [10] and Dogru [5], respectively.On the other hand, a different kind of semi-symmetric nonmetric connection (namely, semi-symmetric recurrent-metric connection) was defined and extensively studied by Andonie and Smaranda [3]; Liang [8].Considering these aspects, we are motivated to study submanifolds of a Riemannian manifold with the semi-symmetric recurrent-metric connection mentioned in [3,8].More precisely, in Section 2, a general description of Riemannian manifold and its submanifold is given and then a semi-symmetric recurrent-metric connection is defined.In Section 3, we show that the induced connection on a submanifold of a Riemannian manifold with the semi-symmetric recurrent-metric connection is also a semi-symmetric recurrent-metric connection.And then the Gauss, the Codazzi-Mainardi and the Ricci equations for such a connection have been derived.We also consider the totally geodesic, totally umbilical and minimal submanifolds of a Riemannian manifold with the semi-symmetric recurrentmetric connection.Finally, a concrete example of submanifold of a Riemannian manifold with the semi-symmetric recurrent-metric connection is given.

Preliminaries
Let M be an n-dimensional submanifold of an (n + l)-dimensional Riemannian manifold M .From now on, g refers to the Riemannian metric tensor on M as well as the induced one on M .Also, in the sequel, X, Y , Z, W denote the vector fields on M ; X, Y, Z, W denote the vector fields tangent to M .The formulas of Gauss and Weingarten are given by and respectively, where η is a normal vector field of M in M and ∇ is the induced Riemannian connection on M from the Riemannian connection ∇ on M and ∇ ⊥ is a (metric) connection in the normal bundle T (M ) ⊥ with respect to the fibre metric induced from g [7].Note that the second fundamental form h is related to the shape operator A η by If h = 0, then M is said to be totally geodesic.The mean curvature vector H of M is given by M is said to be minimal if H=0 ; M is said to be totally umbilical if h(X, Y ) = g(X, Y )H.The covariant derivative of h is defined by The connection ∇ is called the van der Waerden-Bortolotti connection of M [4].In [13], a linear connection ´ ∇ on a Riemannian manifold M is given by where u 1 and u 2 are 1-forms associated with the vector fields U 1 and U 2 on M by u 1 ( X) = g(U 1 , X) and u 2 ( X) = g(U 2 , X), respectively.Using (2.5), the Furthermore, using (2.5), we get Therefore, the linear connection ´ ∇ defined by (2.5) is adequate to be called a semi-symmetric recurrent-metric connection [13].From now on, for the sake of simplicity, a semi-symmetric recurrent-metric connection is briefly denoted by a SSRM connection.We define the curvature tensor of type (1,3) Also, the curvature tensor of type (0,4 (2.9)

Submanifolds of a Riemannian Manifold with a Semi-Symmetric Recurrent-Metric Connection
Let ∇ be the induced connection on M from the SSRM connection ´ ∇ on M by the equation which may be called the formula of Gauss with respect to the SSRM connection ´ where h is a normal bundle valued tensor of type (0,2).Taking account of (2.1), (2.5) and (3.10), we have ∇X Here we denote by U ⊤ i and U ⊥ i the tangential and normal components of U i (i = 1, 2), respectively.It follows from (3.11) and the properties of a Riemannian connection ∇ that and Therefore, we obtain the following: ´ ∇.
Proof.In view of (3.12) and (3.15), we have Comparing the above relation with (3.12), we conclude that a totally umbilical M is also totally umbilical with respect to ´ ∇, and vice versa.On the other hand, let us assume that U 1 and U 2 are tangent to M .Then we have from (3.12) h(X, Y ) = h(X, Y ).
Taking account of the above identity and (3.15), we obtain that (i), (ii), (iii) and (iv) hold true.This completes the proof of theorem 3.2.
Let η be a normal vector field of M in M .Taking account of (2.5), we get which yields from (2.2) ´ From (3.17), we can define a tensor of type (1,1) on M as follows: Taking (3.17) and (3.18) into account, we have ´ which may be called the formula of Weingarten with respect to the SSRM connection ´ ∇.Now we obtain the following:

a submanifold of a Riemannian manifold M with the SSRM connection
´ ∇, then for the unit normal vector field η of M in M , the principal directions and the principal directions with respect to the SSRM connection ´ ∇ coincide.Moreover, if the associated vector field U 1 and U 2 are tangent to M , then the principal curvatures are equal to the principal curvatures with respect to the SSRM connection ´ ∇.
Proof.Taking account of (3.18), we conclude that the principal directions and the principal directions with respect to the SSRM connection ´ ∇ coincide.On the other hand, let us assume that U 1 and U 2 are tangent to M .Then we have from (3.18) Áη = A η , which implies that the principal curvatures are equal to the principal curvatures with respect to ´ ∇.This completes the proof of theorem 3.3.

Theorem 3.4. Let M be a submanifold of a Riemannian manifold M with the SSRM connection
´ ∇.Then the shape operators are simultaneously diagonalizable if and only if the shape operators with respect to the SSRM connection ´ ∇ are simultaneously diagonalizable.
Proof.For the unit normal vector fields η, µ of M in M , we have from (3.18) which gives the required result.
Let Ŕ be the curvature tensor with respect to the induced SSRM connection ∇ on a submanifold M .More precisely, From (2.3), (3.12) and (3.18), it follows that which may be called the Gauss equation with respect to the SSRM connection.
The manifold M (resp.M ) with a SSRM connection ´ ∇ (resp.∇) is said to be ´ ∇-flat (resp.∇-flat) if the curvature tensor ´ R (resp.Ŕ) of ´ ∇ (resp.∇) vanishes.Now we can state the following: Theorem 3.5.Let M be a ´ ∇-flat manifold.If M is totally geodesic and the associated vector fields U 1 , U 2 are tangent to M , then M is a ∇-flat manifold.
Concerning the sectional curvature with respect to the SSRM connection, we have the following: Theorem 3.6.Let M be a submanifold of a Riemannian M with the SSRM connection ´ ∇. (i) If the associated vector fields U 1 , U 2 are tangent to M , then for orthonormal tangent vector fields X, Y on M , we have (ii) Moreover, if α is a geodesic curve of M which lies in M and X is the unit tangent vector field of α in M , then we have Proof.Suppose that X = W , Y = Z are orthonormal tangent vector fields on M .Then we have from (3.21) Let α be a geodesic in M which lies in M and X be a unit tangent vector field of α in M .Then we have from (2.1) If we assume ∇ X Y = 0, then we have from (2.1) This completes the proof of theorem 3.6.
In view of (3.13), (3.18) and (3.20), the normal component of ´ R(X, Y )Z is obtained as follows: which may be called the Codazzi-Mainardi equation with respect to the SSRM connection.Here the connection ´ ∇ is the van der Waerden-Bortolotti connection with respect to SSRM connection defined by For unit normal vector fields η, µ of M in M , we have from (3.10), (3.16) and ´ ), which yields from (2.1), (2.2), (2.3), (3.12) and ( 3 2 )g(η, µ), which may be called the Ricci equation with respect to the SSRM connection.
Example.Let T n be a torus embedded in R 2n as follows:
M is minimal if and only if M is minimal with respect to the SSRM connection ; M is said to be totally umbilical with respect to the SSRM connection ´∇ if h(X, Y ) = g(X,Y ) H. Now we can state the following: Theorem 3.2.Let M be a submanifold of a Riemannian manifold M with the SSRM connection ´ ∇.Then M is totally umbilical if and only if M is totally umbilical with respect to the SSRM connection ´ ∇.Furthermore, if the associated vector fields U 1 and U 2 are tangent to M , then: (i) the second fundamental form h of M and the second fundamental form