On Submanifolds in a Riemannian Manifold with a Semi-Symmetric Non-Metric Connection

In this paper, we study submanifolds in a Riemannian manifold with a semi-symmetric non-metric connection. We prove that the induced connection on a submanifold is also semi-symmetric non-metric connection. We consider the total geodesicness and minimality of a submanifold with respect to the semi-symmetric non-metric connection. We obtain the Gauss, Cadazzi, and Ricci equations for submanifolds with respect to the semi-symmetric non-metric connection.


Introduction
In 1924, Friedmann and Schouten [1] introduced the idea of semi-symmetric connection on a differentiable manifold.A linear connection ∇ on a differentiable manifold M is said to be a semi-symmetric connection if the torsion T of the connection ∇ satisfies where π is a 1-form.In 1932, Hayden [2] introduced the notion of a semi-symmetric metric connection on a Riemannian manifold ( M, g).A semi-symmetric connection ∇ is said to be a semi-symmetric metric connection if ∇g = 0. (2) Yano [3] studied some properties of a Riemannian manifold endowed with a semi-symmetric metric connection.Submanifolds of a Riemannian manifold with a semi-symmetric metric connection were studied by Nakao [4].
After a long gap, the study of a semi-symmetric connection ∇ satisfying ∇g = 0 was initiated by Prvanovic [5] with the name pseudo-metric semi-symmetric connection, and was just followed by Smaranda and Andonie [6].
A semi-symmetric connection ∇ is said to be a semi-symmetric non-metric connection if it satisfies the condition Equation (3).
In 1992, Agashe and Chafle [7] introduced a semi-symmetric non-metric connection on a Riemannian manifold ( M, g) given by where ∇ is the Levi-Civita connection of ( M, g) and π is a 1-form.Agashe and Chafle [8] studied submanifolds of a Riemannian manifold with this semi-symmetric non-metric connection.In 2000, Sengupta, De, and Binh [9] gave another type of semi-symmetric non-metric connection.Özgür [10] studied properties of submanifolds of a Riemannian manifold with this semi-symmetric non-metric connection.Recently, De, Han, and Zhao [11] introduced a new type of semi-symmetric non-metric connection which is given by where a and b are two non-zero real numbers and ω is a 1-form.They proved the existence of this new type of linear connection and studied a Riemannian manifold admitting this type of semi-symmetric non-metric connection in [11].
Motivated by [8] and [10], we have studied submanifolds of a Riemannian manifold endowed with the semi-symmetric non-metric connection defined by Equation ( 4) in this paper.The paper has been organized as follows: In Section 2, we give some properties of the semi-symmetric non-metric connection; In Section 3, we consider a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection and show that the induced connection on the submanifold is also a semi-symmetric non-metric connection.We also consider the total geodesicness and minimality of a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection; In section 4, we deduce the Gauss, Codazzi, and Ricci equations with respect to the semi-symmetric non-metric connection.Using this Gauss equation, we give the relation between the sectional curvatures with respect to the semi-symmetric non-metric connection of a Riemannian manifold and a submanifold, which is analogous to Synger's inequality [12].Finally, we consider these fundamental equations of a submanifold in a space form with constant curvature with the semi-symmetric non-metric connection.

Preliminaries
Let M be an (n + d)-dimensional Riemannian manifold with a Riemannian metric g and ∇ be the Levi-Civita connection of ( M, g).De, Han, and Zhao [11] defined a special type of linear connection on where a and b are two non-zero real numbers and ω is a 1-form on M. Denote by U = ω , i.e., the vector field U is defined by ω( X) = g( X, U) for all X ∈ X ( M), X ( M) is the set of all differentiable vector fields on M. By Equation ( 5), the torsion tensor ˇ T with respect to the connection ˇ ∇ is given by where π( X) = (b − a)ω( X) is a 1-form.Therefore, the connection ˇ ∇ is a semi-symmetric connection.Additionally, Hence, the semi-symmetric connection ˇ ∇ defined by Equation ( 5) is a semi-symmetric non-metric connection.
Analogous to the definition of the curvature tensor R of M with respect to the Levi-Civita connection ∇, we define the curvature tensor ˇ R of M with respect to the semi-symmetric non-metric where X, Y, Z ∈ X ( M).
Using Equations ( 5) and ( 6), we have The Riemannian Christoffel tensors of the connections ∇ and ˇ ∇ are defined by respectively.

Submanifolds of a Riemannian Manifold with the Semi-Symmetric Non-Metric Connection ˇ ∇
Let M be an n-dimensional submanifold of an (n + d)-dimensional Riemannian manifold with the semi-symmetric non-metric connection ˇ ∇.We decompose the vector field U on M uniquely into their tangent and normal components U , U ⊥ .
The Gauss formula for the submanifold M with respect to the Levi-Civita connection ∇ is given by where h is the second fundamental form of M in M.
For the second fundament form h, the covariant of h is defined by Then, ∇h is a normal bundle valued tensor of type (0, 3) and is called the third fundamental form of M. ∇ is called the van der Waerden-Bortolotti connection of M; i.e., ∇ is the connection in TM ⊕ T ⊥ M built with ∇ and ∇ ⊥ .
Let ∇ be the induced connection from the semi-symmetric non-metric connection ˇ ∇.We define where ȟ is a (1, 2)-tensor field in T ⊥ M, the normal part of M. The Equation ( 10) may be called the Gauss formula for M with respect to the semi-symmetric non-metric connection ˇ ∇.Using Equations ( 5), (8), and (10), we have Comparing the tangential and normal parts of Equation (11), we obtain From Equation ( 12), we have where Ť is the torsion tensor of the connection ∇ on M.Moreover, using Equation ( 12), we have In view of Equations ( 12), ( 14), and (15), we can state the following theorem: Theorem 1.The induced connection ∇ on a submanifold of a Riemannian manifold endowed with the semi-symmetric non-metric connection ˇ ∇ is also a semi-symmetric non-metric connection.
If ȟ(X, Y) = 0 for all X, Y ∈ X (M), then M is called totally geodesic with respect to the semi-symmetric non-metric connection.Let {e 1 , • • • , e n } be an orthonormal basis of the tangent space of M. We define the mean curvature vector Ȟ of M with respect to the semi-symmetric non-metric connection by From Equation (13) we know that where H is the mean curvature vector of the submanifold M. If Ȟ = 0, then M is called minimal with respect to the semi-symmetric non-metric connection.From Equations ( 13) and (17), we have the following result: Theorem 2. Let M be an n-dimensional submanifold of an (n + d)-dimensional Riemannian manifold M with the semi-symmetric non-metric connection ˇ ∇.Then, (1) M is totally geodesic with respect to the semi-symmetric non-metric connection if and only if M is totally geodesic with respect to the Levi-Civita connection.
(2) M is minimal with respect to the semi-symmetric non-metric connection if and only if M is minimal with respect to the Levi-Civita connection.
Let ξ be a normal vector field on M. From Equation (5), we have It is well known that the Weingarten formula for a submanifold of a Riemannian manifold is given by where A ξ is the shape operator of M in the direction of ξ.
Then, Equation (20) turns into Equation ( 22) is called the Weingarten formula for M with respect to the semi-symmetric non-metric connection.
Since A ξ is symmetric, it is easy to verify that where and ξ, η are normal vector fields on M.
From Equations ( 21) and ( 23), we can also obtain the following theorems: Theorem 3. Principal directions of the unit normal vector ξ with respect to the Levi-Civita connection ∇ and the semi-symmetric non-metric connection ˇ ∇, and the principle curvatures are equal if and only if ξ is orthogonal to U ⊥ .Theorem 4. Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection ˇ ∇.Then, the shape operators with respect to the semi-symmetric non-metric connection are simultaneously diagonalizable if and only if the shape operators with respect to the Levi-Civita connection are simultaneously diagonalizable.

Gauss, Codazzi, and Ricci Equations with Respect to the Semi-Symmetric Non-Metric Connection
We denote the curvature tensor of a submanifold M of a Riemannian manifold M with respect to the induced semi-symmetric non-metric connection ∇ and the induced Levi-Civita connection ∇ by respectively, where X, Y, Z ∈ X (M).
Theorem 5. Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection ∇.Then, for all X, Y, Z, W ∈ X (M), we have Here Equation ( 25) is called the Gauss equation for the submanifold M with respect to the semi-symmetric non-metric connection.
Proof.From Equations ( 10) and (20), we have Using Equations ( 24), ( 26)-( 28), we obtain Recalling that if π ∈ T p M is a 2-dimensional subspace of T p M spanned by an orthonormal base {X, Y}, we define the sectional curvature Ǩ(π) with respect to the semi-symmetric non-metric connection as Ř(X, Y, Y, X).Let ˇ K(π) denote the corresponding sectional curvature in M. As an application of the Gauss Equation (25), we can obtain the following Synger's inequality with respect to the semi-symmetric non-metric connection.

Corollary 1.
Let M be a submanifold of a Riemannian manifold M with the semi-symmetric non-metric connection ˇ ∇ and γ be a geodesic in M which lies in M, and T be a unit tangent vector field of γ. π is a subspace of the tangent space T p M spanned by {X, T}.Then, if X is a unit tangent vector field on M which is parallel along γ and orthogonal to T, then the equality of (1) holds if and only if X is parallel along γ in M.
Proof.(1) Let γ be a geodesic in M which lies in M and T be a unit tangent vector field of γ.Then, we have Let π be a subspace of the tangent space T p M spanned by an orthonormal base {X, T}.Applying the Gauss Equation ( 25) and h = ȟ, we obtain = Ř(X, T, T, X) − g(h(X, X), h(T, T)) + g(h(X, T), (X, T)) + bω(h(T, T)) = Ǩ(π) + g(h(X, T), (X, T)) (2) If X be parallel along γ, we have ∇ T X = 0. Thus, we have Then, the equality of Equation (31) holds if and only if h(X, T) = 0; i.e., ∇ T X = 0. Theorem 6.Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection ∇.Then, for all X, Y, Z ∈ X (M), we have where 32) is called the Codazzi equation with respect to the semi-symmetric non-metric connection.
Proof.From Equation (29), the normal component of ˇ R(X, Y)Z is given by Remark 1. ∇ is the connection in TM ⊕ T ⊥ M built with ∇ and ∇ ⊥ .It may be called the van der Waerden-Bortolotti connection with respect to the semi-symmetric non-metric connection.
Theorem 7. Let M be a submanifold of a Riemannian manifold with the semi-symmetric non-metric connection ∇.Then, for all X, Y ∈ X (M) and normal vector fields ξ, µ on M, we have Equation ( 33) is called the Ricci equation for the submanifold M with respect to the semi-symmetric non-metric connection.
Proof.From Equations ( 10) and ( 22), we get Using Equations ( 34)-(36), we have In view of Equations ( 10), (13), and (21), the above equation turns into It will be useful to examine the form of our fundamental equations with respect to the semi-symmetric non-metric connection when the ambient space M has constant curvature.Now, assume that M is an (n + d)-dimensional space form of constant curvature C with the semi-symmetric non-metric connection ˇ ∇.Let M be a submanifold of M.Then, from Equation ( 7) we have where X, Y, Z ∈ X (M).
Hence from Equation (25) we know that the Gauss equation becomes From Equation (37) we know Since M is a space form of constant C, it follows that R(X, Y, ξ, µ) = 0. On the other hand, from Equation (37) we have Then, using Equations ( 33) and (38), we obtain that the Ricci equation becomes Using Equations ( 23) and (39), we can state the following result: Corollary 2. Let M be a submanifold of a space form of constant curvature with the semi-symmetric non-metric connection ˇ ∇.Then, the normal connection ∇ ⊥ is flat if and only if all second fundamental tensors with respect to the Levi-Civita connection and the semi-symmetric non-metric connection are simultaneously diagonalizable.
Example.Let T 2 : S 1 (1) × S 1 (1) ∈ R 4 be a torus embedded in R 4 defined by For p = (cos u, sin u, cos v, sin v), T P (T 2 ) is spanned by  Equations ( 45) and ( 46) show that the induced connection ∇ is also a semi-symmetric non-metric connection.