We study hypersurfaces isometrically immersed in a trans-S-manifolds in order to find out under what conditions they could inherit the structure of the ambient manifold and so, to obtain new examples of such trans-S-manifolds. Mainly, we investigate this situation depending the behaviour of the second fundamental form of the immersion.

Recently, B.-Y. Chen discovered a technique to find the relation between second fundamental form and the warping function of warped product submanifolds. In this paper, we extend our further study of [24] by giving non-trivial examples of warped product pointwise hemi-slant submanifolds. Finally, we establish a sharp estimation for the squared norm of the second fundamental form ||h||2 in terms of the warping function f. The equality case is also investigated.

In this paper, we would like to give an answer to \textbf{Problem 1} below issued firstly in [J. Mao, Eigenvalue estimation and some results on finite topological type, Ph.D. thesis, IST-UTL, 2013]. In fact, by imposing some conditions on the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced mean curvature flow considered here, we can obtain that the first eigenvalues of the Laplace and the $p$-Laplace operators are monotonic under this flow. Surprisingly, during this process, we get an interesting byproduct, that is, without any complicated constraint, we can give lower bounds for the first nonzero closed eigenvalue of the Laplacian provided additionally the second fundamental form of the initial hypersurface satisfies a pinching condition.

Recently, wehave discussed the warped product pseudo-slant submanifolds of the typeM?xfM? of Kenmotsu manifolds. In this paper, we study other type of warped product pseudo-slant submanifolds by reversing these two factors in Kenmotsu manifolds. The existence of such warped product immersions is proved by a characterization. Also, we provide an example of warped product pseudo-slant submanifolds. Finally, we establish a sharp estimation such as ||h||2?2pcos2?(||??(ln f)||2-1) for the squared norm of the second fundamental form khk2, in terms of the warping function f, where ??(ln f) is the gradient vector of the function ln f. The equality case is also discussed.

Abstract. We study the geometry of r-lightlike submanifolds M of asemi-Riemannian manifold M with a semi-symmetric non-metric connec-tion subject to the conditions; (a) the screen distribution of M is to-tally geodesic in M, and (b) at least one among the r-th lightlike secondfundamental forms is parallel with respect to the induced connection ofM. The main result is a classi cation theorem for irrotational r-lightlikesubmanifold of a semi-Riemannian manifold of index r admitting a semi-symmetric non-metric connection. 1. IntroductionThe geometry of lightlike submanifolds is used in mathematical physics, inparticular, in general relativity since lightlike submanifolds produce models ofdi erent types of horizons (event horizons, Cauchy’s horizons, Kruskal’s hori-zons). The universe can be represented as a four dimensional Lorentz subman-ifold (spacetime) embedded in an (n+ 4)-dimensional semi-Riemannian mani-fold. Lightlike hypersurfaces are also studied in the theory of electromagnetism[1]. Thus, large number of applications but limited information available, moti-vated us to do research on this subject matter. Duggal-Bejancu [1] and Kupeli[2] developed the general theory of degenerate (lightlike) submanifolds. Theyconstructed a transversal vector bundle of lightlike submanifold and investi-gated various properties of these manifolds. Duggal-Jin [3] studied totallyumbilical lightlike submanifold of a semi-Riemannian manifold. Ageshe andChae [4] introduced the notion of a semi-symmetric non-metric connection ona Riemannian manifold. Ya˘sar, C˘oken and Yucesan [5] and Jin [6] studied light-like hypersurfaces in semi-Riemannian manifolds admitting a semi-symmetricnon-metric connections. The geometry of half lightlike submanifolds of a semi-Riemannian manifold with semi-symmetric non-metric connection was studied

Consider a family of smooth immersions F(; t) : Mn ! Rn+k of submanifolds in Rn+k moving by mean curvature ow @F @t = ~H, where ~H is the mean curvature vector for the evolving submanifold. We prove that for any n 2 and k 1, the ow starting from a closed submanifold with small L2-norm of the traceless second fundamental form contracts to a round point in nite time, and the corresponding normalized ow converges exponentially in the C1-topology, to an n-sphere in some subspace Rn+1 of Rn+k.

Recently, we have researched certain twisted product CR-submanifolds in a Kaehler manifold and some inequalities of the second fundamental form of these submanifolds [11]. We consider here two kinds of twisted product CR-submanifolds (the first and the second kind) in a locally conformal Kaehler manifold. In these submanifolds, we give inequalities of the second fundamental form (see Theorems 5.1 and 5.2) and consider the equality case of these.

In this paper, we initiate the study of ${\mathcal{P}} R$-warped products in para-K a hler manifolds and prove some fundamental results on such submanifolds. In particular, we establish a general optimal inequality for ${\mathcal{P}}R$-warped products in para-K a hler manifolds involving only the warping function and the second fundamental form. Moreover, we completely classify ${\mathcal{P}} R$-warped products in the flat para-K a hler manifold with least codimension which satisfy the equality case of the inequality. Our results provide an answer to the Open Problem (3) proposed in [19, Section 5].

In this paper we investigate $(n+1)(n \geq 5)$-dimensional contact $CR$-submanifolds $M$ of $(n-1)$ contact $CR$-dimension in a $(2m+1)$-dimensional unit sphere $S^{2m+1}$ which satisfy the condition $h(FX,Y)-h(X,FY)=g(FX, Y)\zeta $ for a normal vector field $\zeta$ to $M$, where $h$ and $F$ denote the second fundamental form and a skew-symmetric endomorphism (defined by (2.3)) acting on tangent space of $M$, respectively..

A three-dimensional homogeneous Lorentzian manifold is either symmetric or locally isometric to a Lie group equipped with a left-invariant Lorentzian metric \cite{C1}. We completely classify surfaces with parallel second fundamental form in all non-symmetric homogeneous Lorentzian three-manifolds. Interesting differences arise with respect to the Riemannian case studied in [11, 12].

LetM1 be a closed submanifold isometrically immersed in a unit sphereSn+p. Denote byR, H andS, the normalized scalar curvature, the mean curvature, and the square of the length of the second fundamental form ofM1, respectively. SupposeR is constant and ≥1. We study the pinching problem onS and prove a rigidity theorem forM1 immersed inSn+p with parallel normalized mean curvature vector field. Whenn≥8 or,n=7 andp≤2, the pinching constant is best.

A surface x: M → Sn is called a Willmore surface if it is a criticalsurface of the Willmore functional ∫M (S − 2H2)dv, where H isthe mean curvature and S is the square of the length of the secondfundamental form. It is well known that any minimal surface is aWillmore surface. The first nonminimal example of a flat Willmoresurface in higher codimension was obtained by Ejiri. This example whichcan be viewed as a tensor product immersion of S1(1) and a particularsmall circle in S2(1), and therefore is contained in S5(1) gives anegative answer to a question by Weiner. In this paper we generalize theabove mentioned example by investigating Willmore surfaces in Sn(1)which can be obtained as a tensor product immersion of two curves. We inparticular show that in this case too, one of the curves has to beS1(1), whereas the other one is contained either in S2(1) or in S3(1). In the first case, we explicitly determine the immersion interms of elliptic functions, thus constructing infinetely many newnonminimal flat Willmore surfaces in S5. Also in the latter casewe explicitly include examples.

A formulation of the ECSK (Einstein-Cartan-Sciama-Kibble) theory with a Dirac spinor is given in terms of differential forms with values in exterior vector bundles associated with a fixed principalSL(2, ℂ)-bundle over a 4-manifold. In particular, tetrad fields are represented as soldering forms. In this setting, both the scalar curvature (Einstein-Hilbert) action density and the Dirac action density are well-defined polynomial functions of the soldering form and an independentSL(2,ℂ)-connection form. Thus, these densities are defined even where the tetrad field is degenerate (e.g. when fluctuations in the gravitational field are large). A careful analysis of the initial-value problem (in terms of an evolving triad field, SU(2)-connection, second-fundamental form and spinor field) reveals a first-order hyperbolic system of 27 evolution equations (not including the 8 evolution equations for the Dirac spinor) and 16 constraints. There are 10 conservation equations (due to local Poincare invariance) which team up with some of the evolution equations to guarantee that the 16 constraints are preserved under the evolution.

Homotopien von Untermannigfaltigkeiten mit nicht ausgearteter zweiter Fundamentalform