sectional curvature of submanifold

Abstract. manifold of constant holomorphic sectional curvature c, and let M be Received by the editors February 26, 1973. Entropy 2018, 20, 529 3 of 15 of the extrinsic normalized d-Casorati curvatures and the intrinsic scalar curvature of statistical submanifolds in Kenmotsu statistical manifolds of constant f-sectional curvature.The methodology is based on a constrained extremum problem. 1 Smoothing the Conic Metric PDF The curvature of a Hessian metric - UCLA Mathematics Recall that for a pseudo-Riemannian submanifold (M;g) of . Since sectional curvature offers a lot of information concerning the intrinsic geometry of Riemannian manifolds, therefore in this paper, we obtain the expressions for sectional curvature, holomorphic sectional curvature, and holomorphic bisectional curvature of a -lightlike submanifold of an indefinite Kaehler manifold. S. K. Hui et al. Moreover, we prove that the equality cases of the . SOBOLEV METRICS ON DIFFEOMORPHISM GROUPS AND THE DERIVED GEOMETRY OF SPACES OF SUBMANIFOLDS . These estimates depend on the sectional curvature of M , the Weingarten map of P and the radius of the tube. PDF Chapter 14 Curvature in Riemannian Manifolds sectional curvature c. Consider a one-parameter family of smooth immer- . PDF Riemannian Manifolds With Positive Sectional Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit . Note that for any x∈ Mthere is a neighbourhood Uof . The sectional curvature is indeed a simpler object, and it turns out that the curvature tensor can be recovered from it. Let Mbe a compact 3-dimensional totally real submanifold of S6. Also, let h be the second fundamental form and R the Riemann curvature tensor of M. We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite Sasakian manifold and obtain some characterization theorems on holomorphic sectional and holomorphic bisectional curvature. (The same goes for a more general concept, holomorphic bisectional curvature.) fold Nn with nonnegative sectional curvature. Lalonde and Polterovich proved that if $\phi $ is a bounded compactly supported symplectomorphism of $M, L\subset M$ is a closed Lagrangian submanifold admitting a Riemannian metric with non-positive sectional curvature, and the inclusion of L in M induces an injection on fundamental groups, then $\phi (L)\cap L\ne \emptyset $ [38 . The symplectic sectional curvature of the symplectic subspace spanned by X and JX is simply the product of the (para-)holomorphic sectional curvature of the span of X and JX with the quadratic form on span { X, J X } determined by g. To make this precise, define (4.4) s ( θ) = { sin. adshelp[at]cfa.harvard.edu The ADS is operated by the Smithsonian Astrophysical Observatory under NASA Cooperative Agreement NNX16AC86A negative sectional curvature, . We establish some inequalities involving the normalized δ-Casorati curvatures (extrinsic invariants) and the scalar curvature (intrinsic invariant). For this purpose, the bounded sectional curvature is introduced and some special submanifolds of r-lightlike submanifolds of a semi-Riemannian manifold are investigated. In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. If the Ricci curvature . If: a) the sectional curvature is non-positive for all tangent planes of M that contain tangent vectors to geodesics of M that are perpendicular 1. In Riemannian geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds with dimension greater than 1. In 2011, K. Liu, H. Xu, F. Ye and . Then we give basic equations and deﬂnitions for a submanifolds. As a consequence we also prove a rigidity result for the Simon-Smith minimal spheres. Introduction Let M be a connected real n-dimensional submanifold of real codimension Ricci curvature, the scalar curvature and the sectional curvature of the submanifold. Then M3 is either Sasakian, ﬂat or of constant ˘-sectional curvature r<1 and constant ˚-sectional curvature r. 3 Anti-invariant submanifold of N( )-contact metric manifold We denote the Ricci curvature at X by Ric(X) = K12 + ¢¢¢ + K1n; where Kij denotes the sectional curvature of the 2-plane section spanned by ei;ej. Complex submanifolds Let M be a submanifold of a Riemannian manifold N with metric tensor g. Denote by R M and R N the Riemannian curvature tensors of M and N and by a the second fundamental form of M in N. Then the Gauss-Codazzi equa- It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ p as a tangent . In [21,22], Opozda presented a statistical sectional curvature on a statistical manifold. Sorted by: Results 1 - 6 of 6. +K1k, (2.9) where Kijdenotes the sectional curvature of the 2-plane section spanned by ei, ej.We simply called such a curvature a k-Ricci curvature. Extrinsic curvature of submanifolds In this lecture we deﬁne the extrinsic curvature of submanifolds in Euclidean space. I would also be interested in hearing about the variation of the problem with constant sectional curvature metrics. In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. 2.2 Sectional curvature We will now talk about the sectional curvature, an important simpli cation of the Rieman-nian Curvature Tensor as it completely determines it. dimensional oriented closed C-totally real submanifold Mwith parallel mean curvature vector hin a (2m+ 1)-dimensional closed -Sasakian space form M~(c) of constant '-sectional curvature cwith 0 <c , n 2 and if a tensor ˚related to hand the second fundamental form satis es a certain inequality. 0), where H = 1 n trace h is the mean curvature vector of M in N. For a totally umbilical submanifold M the shape The scalar curvature τof the k-plane section Lis given by τ(L)= 1≤i<j≤k Kij. AMS (MOS) subject classifications (1970). Let (Mn,g,X) be closed with positive weighted sectional curvature. We give some estimates for the supremun of the sectional curvature of a submanifold N of a Riemannian or Kaehlerian manifold M such that N is contained in a tube about a submanifold P of M . in dimension 9, which do not admit positive curvature. In a seminal paper published in the early 1990s, Chen [] established a sharp inequality for a submanifold in a real space form using the scalar curvature and the sectional curvature, both being intrinsic invariants, and squared mean curvature, the main extrinsic invariant, initiating the theory of δ-invariants or the so-called Chen invariants; this turned out to be one of the most interesting . Tools. If the sectional curvature Kof Msatis es 1 16 <K 1; (1) then Mis a totally geodesic submanifold (i.e. Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks (0) by M Micheli, P W Michor, D B Mumford Venue: SIAM J. Given any two vectors u,v 2 T pM,recallbyCauchy-Schwarz that In 15 , Kulkarni . In addition, the equality case is likewise discussed. Scalar curvature, sectional curvature, Kaehler submanifold. In particular there exist exotic spheres, e.g. Key words and phrases. Sectional curvature of 3-manifolds In section (X, |8) will always dénote a three dimensional Lorentzian this manifold and thus hâve signature h). In this article, we consider statistical submanifolds of Kenmotsu statistical manifolds of constant ϕ-sectional curvature. The main result of this paper is an expression of the flag curvature of a submanifold of a Randers-Minkowski space $({\mathscr V},F)$ in terms of invariants related to its Zermelo data $(h,W)$.. In section 2, we recall the deﬂnitions of the Ricci curvature, the k-Ricci curvature, the scalar curvature, and the normalized scalar cur-vature. The squared mean curvature is the most impor-tant, among the extrinsic invariants of a submanifold, and the Ricci curvature, the sectional curvature, k-invariant and the scalar curvature are well-known among its [4] Let M3 be a contact metric manifold on which Q˚= ˚Q. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. 1. Section ifold to be a geodesic hypersphere in CP(n+p)/2 in terms with sectional curvature. For such submanifold, we investigate curvature properties. to the case of statistical submanifolds in a statistical manifold of constant curvature, obtaining a lower bound for the Ricci curvature of the dual connections. For example, every complex submanifold of C n (with the induced metric from C n) has holomorphic sectional curvature ≤ 0. We obtain the expressions for sectional curvature, holomorphic sectional curvature and holomorphic bisectional curvature of a GCR-lightlike submanifold of an indefinite Sasakian manifold and obtain some characterization theorems on holomorphic sectional and holomorphic bisectional curvature. Moreover, we prove that the equality cases of the inequalities . We fix some point peX and investi- gate the sectional curvature at p. We assume that local coordinates hâve been K& chosen near p such that the metric tensor is represented by diag 1, 1, +1) at p. In [5], B.Y. 1 Introduction. Surprisingly, for positive curvature one has in addition only the classical obstructions: (4c) the complex projective space of constant holomrophic sectional curvature 4c, the quaternionic projective space of constant quaternionic sectional curvature 4cand the Cayley plane of constant octonian sectional curvature 4c, and an n-dimensional manifold of positive constant curvature 4. K 1 on M). θ if ϵ = − 1, sinh. Section The objective of this paper is to achieve the inequality for Ricci curvature of a contact CR-warped product submanifold isometrically immersed in a generalized Sasakian space form admitting a nearly Sasakian structure in the expressions of the squared norm of mean curvature vector and warping function. DENSITY OF A MINIMAL SUBMANIFOLD AND TOTAL CURVATURE OF ITS BOUNDARY Jaigyoung Choe and Robert Gulliver Abstract Given a piecewise smooth submanifold n 1 ˆR mand p2R , we de ne the vision angle p() to be the ( n 1)-dimensional volume of the radial projection of to the unit sphere centered at p. In [21,22], Opozda presented a statistical sectional curvature on a statistical manifold. The Gauss curvature equation is used to prove inequalities relating the sectional curvatures of a submanifold with the corresponding sectional curvature of the ambient manifold and the size of the . In 1979, Kulkarni [1] proved that the sectional curvature of a semi-Riemannian manifold M is unbounded from above and below at each point unless the manifold has constant sectional curvature. Let V be vector space, x;y 2 V jx^yj p . ⁡. Example : S2 ×S2 ⊂ (R3 ×R3 = R6) has nonnegative sectional curvatures but has positive Ricci curvatures. known as Casorati curvature of the submanifold. For the classical solution of the mean curvature ﬂow, most works have been done If (M, g) is a complete connected Riemannian manifold with sectional curvature K ≥ 0, then there exists a compact totally convex, totally geodesic submanifold S whose normal bundle is diffeomorphic to M. (Note that the sectional curvature must be non-negative everywhere, but it does not have to be constant.) The Gauss curvature equation is used to prove inequalities relat- ing the sectional curvatures of a submanifold with the corresponding curvature of the ambient manifold and the size of the second fundamental form. Let M2p+sbe an invariant submanifold of an S-mani-fold N2n+s. In section 2, we recall the deﬂnitions of the Ricci curvature, the k-Ricci curvature, the scalar curvature, and the normalized scalar cur-vature. Introduction Let gbe a complete Riemannian metric on the 2-sphere S2. Chen proved the following Chen-Ricci inequality on Ricci curvature for any n-dimensional submanifold in Riemannian manifold of constant sectional curvature c: Ric(X) • n ¡ 1 4 c . As a result, the statistical curvature tensor fields of M ¯ and M are defined as: Mathematics Subject Classiﬁcation: 53C40, 53C25 Keywords: complex projective space, CR-submanifold, maximal CR-dimension, sectional curvature, minimal real hypersurface 1 Introduction Let M be an n-dimensional CR-submanifold of maximal CR-dimension iso- 2.2 Sectional curvature We will now talk about the sectional curvature, an important simpli cation of the Rieman-nian Curvature Tensor as it completely determines it. (the average sectional curvature of the 2-planes P containingv). Introduction The most symmetric of all Riemannian manifolds (M;g) are the real space forms, i.e. BISECTIONAL CURVATURE 227 A Riemannian manifold with metric tensor g and Ricci tensor S is called an Einstein space if 5 = kg for some constant k. 4. The proof uses the construction of ancient mean curvature ows that ow out of a minimal submanifold. The usual definitions, in general, cannot produce a sectional curvature with regard to dual connections (which, of course, are not metric). Keywords: curvature; lightlike submanifold; semi-Riemannian manifold. M has positive sectional curvature on some 2-plane at every point (Lemmas 4.1 and 5.1). K ij and K¯ ij are the sectional curvatures of the plane sections in Mm+1 and M¯ 2n+1(c) respectively.Let dim M 1 = 2n Let $ R _ {ij} $ denote the Ricci tensor and let $ Q $ be the quadratic form on $ T _ {P} M ^ {n} $ given by $ R _ {ij} $ at $ P \in M ^ {n} $. Lecture 17. An application to compact submanifolds in obtained. Assume Nn has positive k-th Ricci curvature at all points of Mn~\ If Vr is an immersed closed ri>,k)-dimensional minimal submanifold of Nn which is on one side of Mn~\ then Vr is contained This inequality was extended in 2015 by M. E. Aydin et al. The sectional curvature of a Riemannian space $ M ^ {n} $ at $ P $ in the direction of the tangent plane $ \sigma $ is also called the Riemannian curvature. Suppose dimM = 2, then there is only one sectional curvature at each point, which is exactly the well-known Gaussian curvature (exercise): = R 1212 g 11g 22 g2 12: In fact, for Riemannian manifold M of higher dimensions, K(p) is the Gaussian curvature of a 2-dimensional submanifold of Mthat is tangent to p at p. Example. 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