Geometric inequality of warped product semi-slant submanifolds of locally product Riemannian manifolds

Abstract: In the present article, we derive an inequality in terms of slant immersions and well define warping function for the squared norm of second fundamental form for warped product semi-slant submanifold in a locally product Riemannian manifold. Moreover, the equality cases are verified and generalized the inequality for semi-invariant warped products in locally Riemannain product manifold.


Introduction
The notion of warped product manifolds plays very important roles not only in differential geometry but also in general relativity theory in physics. For example, Robertson-Walker space-times, asymptotically flat spacetime, Schwarzschild spacetime, and Reissner-Nordstrom spacetime are warped product manifolds (Hiepko, 1979). The geometry of warped products has a crucial role in differential geometry, as well as physical sciences. Bishop and O'Neill (1969) discovered the concept of warped product manifolds to derive an example of Riemannian manifolds of negative curvature, such manifolds are natural generalizations of Riemannian products manifolds. Therefore, many geometers are studied in Ali and Luarian (2017), Ali, Othman, and Ozel (2015), Ali and Ozel (2017), Ali, Uddin, and Othman (2017), Al-Solamy and Khan (2012) He is also working as principal and co-principal investigator in several ongoing projects of King Khalid University.

PUBLIC INTEREST STATEMENT
The extrinsic geometry of such warped product submanifolds actually is the mathematization that explicates our awareness of different concrete shapes in given ambient spaces and the intrinsic geometry of such warped product submanifolds is proper and Riemannian geometry. The study of warped products from this extrinsic point of view was initiated around the beginning of this century. Since then the study of warped product submanifolds from the extrinsic point of view has become a very active research subject in differential geometry and many nice results on this subject have been obtained by many geometers. In similar, we obtained the relation between the second fundamental form, the main extrinsic invariant, the main intrinsic invariants are the warping function of a warped product semi-slant submanifolds and slant angle. and , Atceken (2008Atceken ( , 2013, Chen (2001), Sahin (2006aSahin ( , 2006bSahin ( , 2006c. It is interesting to see that there exist no warped product semi-slant submanifolds of the forms M ¼ M θ Â f M T and M such that M T and M θ are holomorphic and slant submanifolds, respectively (see Sahin, 2006b). While, Atceken (see examples 3.1 (Atceken, 2008)) has given an example on the existence of warped product semi-slant submanifold of the form M ¼ M θ Â f M T in a locally product Riemannian manifold such that M T and M θ are invariant and slant submanifolds, respectively. Hence, the geometry of warped product submanifolds in a locally product Riemannian manifold is different from the geometry of warped product submanifolds in Kaehler manifold. Therefore, we consider such a warped product semi-slant submanifold as mixed totally geodesic of locally product Riemannian manifold and obtain a geometric inequality for the length of the second fundamental form in terms of slant immersion and warping functions.

Preliminaries
Assume that M be a manifold of dimension m with a tensor field of such that where F is a one-one tensor field and I represent the identity transformation. Thus, M is an almost product manifold with almost product structure F. If an almost product manifold M admits a Riemannian metric g satisfying gðFU; FVÞ ¼ gðU; VÞ; gðFU; VÞ ¼ gðU; FVÞ; (2:2) 2010 Mathematics Subject Classification. 53C40 Primary 53C20 53C42 secondary.
Key words and phrases. Mean curvature, warped products, Riemannian manifolds, semi-slant immersions. For any U; V 2 ΓðT MÞ, where ΓðT MÞ denotes the set of all vector fields of M then M is said to be an almost product Riemannian metric manifold. Denote Ñ the Levi-Civita connection on M with respect to g. If ð Ñ U FÞV ¼ 0, for all U; V 2 ΓðT MÞ, then ð M; gÞ is a locally product Riemannian manifold with Riemannian metric g (see Sahin, 2006a).
Let M be a submanifold of locally product Riemannian manifold M with an induced metric g. If Ñ ? and Ñ are induced Riemannian connections on normal bundle T ? M and tangent bundle TM and of M, respectively, then Gauss and Weingarten formulas are given by For any X 2 ΓðTMÞ, we can write where PUðtNÞ and ωUðfNÞ are tangential and normal components of FUðFNÞ, respectively. The covariant derivatives of the endomorphism F as ΓðT MÞ: (2:6) A submanifold M of a locally product Riemannian manifold M is said to be totally umbilical (and totally geodesic respectively) if hðU; VÞ ¼ gðU; VÞH; & hðU; VÞ ¼ 0; (2:7) for all U; V 2 ΓðTMÞ. Then H is a mean curvature vector of M given by H ¼ 1 n ∑ n i¼1 hðe i ; e i Þ, where n is the dimension of M and e 1 ; e 2 ; Á Á Á ; e n f g is a local orthonormal frame of the tangent vector space Definition 2.1. A submanifold M of a locally product Riemannian manifold M, then for each non zero vector U tangent to M at a point p, the angle θðUÞ between FU and T p M is called a Wirtinger angle of U. Hence, M is said to be a slant submanifold if the Wirtinger angle is constant and it is independent from the choice of U 2 T p M and p 2 M. The holomorphic and totally real submanifolds are slant submanifolds with slant angle θ ¼ 0 and θ ¼ π=2, respectively. A slant submanifold is said to be proper if it is neither holomorphic nor totally real. More generally, a distribution D on M is called a slant distribution if the angle θðXÞ between FX and D x has same value of θ for each x 2 M and a non zero vector X 2 D x .
Thus for a slant submanifold M, a normal bundle T ? M can be expressed as ( 2:8) where ν is an invariant normal bundle with respect to F orthogonal to ωðTMÞ. We recall following result for a slant submanifold of a locally product Riemannian manifold given by H. Li (cf. Li & Li, 2005). As a consequence for a differentiable function φ : M ! R, we have where gradient Ñφ is defined by gðÑφ; XÞ ¼ Xφ, for any X 2 ΓðTMÞ.

Semi-slant submanifolds
Semi-slant submanifolds were described by Papaghiuc (1994). These submanifolds are generalizations of CR-submanifolds with slant angle θ ¼ π=2. 4. Warped product submanifolds with the form M θ Â f M T Let ðM 1 ; g 1 Þ and ðM 2 ; g 2 Þ be two Riemannian manifolds with a f : M 1 ! ð0; 1Þ, a positive differentiable function on M 1 , we define on the product manifold M 1 Â M 2 with metric g ¼ π Ã g 1 þ ðfoπÞγ Ã g 2 , where π and γ are natural projections on M 1 and M 2 . Under these condition the product manifold is called warped product of M 1 and M 2 , it is denoted by M 1 Â f M 2 and f is called warping function. So we have the following lemma Lemma 4.1 Let M ¼ M 1 Â f M 2 be a warped product manifold. Then for any X; Y 2 ΓðTM 1 Þ and Z; W 2 ΓðTM 2 Þ, we have ðiÞ Ñ X Y 2 ΓðTM 1 Þ:  Now, we develop some important lemmas for first type warped product for later use in the inequality and we refer for example to see their existence, Example 4.1 in Atceken (2008). for any Z 2 ΓðTM θ Þ and X; Y 2 ΓðTM T Þ. for any Z 2 ΓðTM θ and X 2 ΓðTM T Þ:

An inequality for semi-slant warped product submanifolds
In this section, we obtain a geometric inequality for a warped product semi-slant submanifold in terms of the second fundamental form and the warping function with mixed totally geodesic submanifold. Now, we describe an orthonormal frame for a semi-slant submanifold, which we shall use in the proof of inequality theorem. and D, respectively. We consider e 1 ; e 2 ; . . . e β ; e βþ1 ¼ Fe 1 ; . . . e 2β ¼ Fe β È É and which are orthonormal frames of D and D θ respectively. Thus the orthonormal frames of the normal sub bundles, ωD θ and invariant sub bundle ν, respectively are fe m þ 1 ¼ . . . e m þ 2α ¼ e 2α ¼ csc θ sec θωPe Ã α g and e m þ 2α þ 1 ; . . . e 2n f g . Since, M is mixed totally geodesic, then we get h k k 2 ¼ hðD; DÞ k k 2 þ hðD θ ; D θ Þ 2 (5:2) Leaving second term and using (2.11) in first term, we obtain r;k¼1 gðhðe r ; e k Þ; e l Þ 2 : The above expression can be written as in the components of ωD θ and ν, then we derive r;k¼1 gðhðe r ; e k Þ; e l Þ 2 : (5:3) We will remove the last term and using the adapted frame for ωD θ , we derive r;k¼1 gðhðe r ; e k Þ; ωe Ã j Þ 2 þ csc 2 θsec 2 θ ∑ α j¼1 ∑ 2β r;k¼1 gðhðe r ; e k Þ; ωPe Ã j Þ 2 : Again using the adapted frame for D and the fact that second fundamental form is symmetric, then we get Then using Lemma 4.2 and Lemma 4.3, we arrive at ðPe Ã j ln f Þgðe r ; e k Þ n o 2 : Thus combining first and second terms and using the property of trigonometric identities in the third and fourth terms, we get ðPe Ã j ln f Þ 2 : Last the above equation can be modified as jjhjj 2 ! 4βfcsc 2 θjjÑ θ ln f jj 2 g þ 2β ∑ q j¼1 ðsec θPe Ã j ln f Þ 2 À 2β ∑ q j¼1 ðe Ã j ln f Þ 2 : From definition of adapted frame for D θ , finally, we obtain h k k 2 ! 4βcsc 2 θ Ñ θ ln f 2 : If the equality holds, from the leaving terms in (5.2) and (5.3), we obtain the following conditions, i.e., M θ is totally geodesic in M and hðD; DÞ & ν. So the equality case holds. It is completed proof of the theorem.

Conclusion remark
If we assume that the slant angle θ ¼ π 2 ; then warped product semi-slant submanifold M θ Â f M T becomes a warped product semi-invariant submanifold of type M ? Â f M T of a locally product Riemannian manifold, in this case, Theorem 5.1 is generalized to the inequality theorem which was obtained by Sahin (2006a). Therefore, we say that Theorem 5.1 in Sahin (2006c) is trivial case of our derived Theorem 5.1, that is