scholarly journals Submanifolds and the length of the second fundamental tensor

1983 ◽  
Vol 34 (1) ◽  
pp. 1-6
Author(s):  
Thomas Hasanis
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Mean Curvature ◽  
Sufficient Condition ◽  
Fundamental Tensor ◽  
The Mean

AbstractA sufficient condition, for a complete submanifold of a Riemannian manifold of positive constant curvature to be umbilical, is given. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

On the non-regularity of tangent sphere bundles

1978 ◽  
Vol 82 (1-2) ◽  
pp. 13-17 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Riemannian Manifold ◽  
Large Class ◽  
Constant Curvature ◽  
Positive Constant ◽  
Contact Structure ◽  
Compact Riemannian Manifold ◽  
Contact Structures ◽  
Sphere Bundle ◽  
Contact Manifolds ◽  
Tangent Sphere

SynopsisClassically the tangent sphere bundles have formed a large class of contact manifolds; their contact structures are not in general regular, however. Specifically we prove that the natural contact structure on the tangent sphere bundle of a compact Riemannian manifold of non-positive constant curvature is not regular.

ON FINSLER MANIFOLDS WHOSE TANGENT BUNDLE HAS THE g-NATURAL METRIC

2011 ◽  
Vol 08 (07) ◽  
pp. 1593-1610 ◽  
Author(s):  
ESMAEIL PEYGHAN ◽  
AKBAR TAYEBI
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Tangent Bundle ◽  
Contact Structure ◽  
Jacobi Operator ◽  
Riemannian Metric ◽  
Finsler Manifolds ◽  
Almost Contact Structure ◽  
Flat Riemannian Manifold

In this paper, we introduce a Riemannian metric [Formula: see text] and a family of framed f-structures on the slit tangent bundle [Formula: see text] of a Finsler manifold Fn = (M, F). Then we prove that there exists an almost contact structure on the tangent bundle, when this structure is restricted to the Finslerian indicatrix. We show that this structure is Sasakian if and only if Fn is of positive constant curvature 1. Finally, we prove that (i) Fn is a locally flat Riemannian manifold if and only if [Formula: see text], (ii) the Jacobi operator [Formula: see text] is zero or commuting if and only if (M, F) have the zero flag curvature.

scholarly journals On pinching theorems for compact pseudo-umbilical submanifold

2012 ◽  
Vol 45 (3) ◽  
pp. 645-654
Author(s):  
Jing Mao ◽  
Shaodong Qin
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Mean Curvature ◽  
Sufficient Conditions ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Parallel Mean Curvature Vector ◽  
Totally Umbilical ◽  
Totally Umbilical Submanifold ◽  
Parallel Mean Curvature

AbstractConsider submanifolds in the nested space. For a compact pseudoumbilical submanifold with parallel mean curvature vector of a Riemannian submanifold with constant curvature immersed in a quasi-constant curvature Riemannian manifold, two sufficient conditions are given to let the pseudo-umbilical submanifold become a totally umbilical submanifold.

scholarly journals Biharmonic submanifolds in manifolds with bounded curvature

2016 ◽  
Vol 27 (11) ◽  
pp. 1650089
Author(s):  
Shun Maeta
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Biharmonic Submanifolds ◽  
Bounded Curvature ◽  
Negative Constant ◽  
The Mean ◽  
Biharmonic Submanifold

We consider a complete biharmonic submanifold [Formula: see text] in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant [Formula: see text]. Assume that the mean curvature is bounded from below by [Formula: see text]. If (i) [Formula: see text], for some [Formula: see text], or (ii) the Ricci curvature of [Formula: see text] is bounded from below, then the mean curvature is [Formula: see text]. Furthermore, if [Formula: see text] is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

scholarly journals Randers manifolds of positive constant curvature

2003 ◽  
Vol 2003 (18) ◽  
pp. 1155-1165 ◽  
Author(s):  
Aurel Bejancu ◽  
Hani Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Constant Curvature ◽  
Positive Constant ◽  
Complete Riemannian Manifold ◽  
Flag Curvature ◽  
Dimensional Sphere ◽  
Randers Metric ◽  
Constant Flag Curvature ◽  
Simply Connected

We prove that any simply connected and complete Riemannian manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it.

scholarly journals Pseudo-Reimannian manifolds endowed with an almost paraf-structure

1985 ◽  
Vol 8 (2) ◽  
pp. 257-266 ◽  
Author(s):  
Vladislav V. Goldberg ◽  
Radu Rosca
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Mean Curvature ◽  
Integral Manifold ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Minimal Hypersurfaces ◽  
Reimannian Manifolds ◽  
The Mean ◽  
Orthogonal Distribution

LetM˜(U,Ω˜,η˜,ξ,g˜)be a pseudo-Riemannian manifold of signature(n+1,n). One defines onM˜an almost cosymplectic paraf-structure and proves that a manifoldM˜endowed with such a structure isξ-Ricci flat and is foliated by minimal hypersurfaces normal toξ, which are of Otsuki's type. Further one considers onM˜a2(n−1)-dimensional involutive distributionP⊥and a recurrent vector fieldV˜. It is proved that the maximal integral manifoldM⊥ofP⊥hasVas the mean curvature vector (up to1/2(n−1)). If the complimentary orthogonal distributionPofP⊥is also involutive, then the whole manifoldM˜is foliate. Different other properties regarding the vector fieldV˜are discussed.

scholarly journals Embedded positive constant r-mean curvature hypersurfaces in Mm × R

2005 ◽  
Vol 77 (2) ◽  
pp. 183-199 ◽  
Author(s):  
Xu Cheng ◽  
Harold Rosenberg
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Positive Constant ◽  
Mean Curvature ◽  
Rotational Symmetry ◽  
Product Manifold ◽  
Height Estimates ◽  
Topological Obstructions ◽  
Complete Surfaces

Let M be an m-dimensional Riemannian manifold with sectional curvature bounded from below. We consider hypersurfaces in the (m + 1)-dimensional product manifold M × R with positive constant r-mean curvature. We obtain height estimates of certain compact vertical graphs in M × R with boundary in M × {0}. We apply this to obtain topological obstructions for the existence of some hypersurfaces. We also discuss the rotational symmetry of some embedded complete surfaces in S² × R of positive constant 2-mean curvature.

scholarly journals Local Well-Posedness of Skew Mean Curvature Flow for Small Data in $$d\ge 4$$ Dimensions

2022 ◽  
Author(s):  
Jiaxi Huang ◽  
Daniel Tataru
Keyword(s):  
Riemannian Manifold ◽  
Evolution Equation ◽  
Sobolev Spaces ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Small Data ◽  
Well Posedness ◽  
The Mean ◽  
Schrödinger Map

AbstractThe skew mean curvature flow is an evolution equation for d dimensional manifolds embedded in $${{\mathbb {R}}}^{d+2}$$ R d + 2 (or more generally, in a Riemannian manifold). It can be viewed as a Schrödinger analogue of the mean curvature flow, or alternatively as a quasilinear version of the Schrödinger Map equation. In this article, we prove small data local well-posedness in low-regularity Sobolev spaces for the skew mean curvature flow in dimension $$d\ge 4$$ d ≥ 4 .

scholarly journals Further geometry of the mean curvature one-form and the normal plane field one-form on a foliated Riemannian manifold

1997 ◽  
Vol 62 (1) ◽  
pp. 46-63 ◽  
Author(s):  
Grant Cairns ◽  
Richard H. Escobales
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Final Section ◽  
General Context ◽  
Topological Properties ◽  
Normal Plane ◽  
Plane Field ◽  
The Mean ◽  
Foliated Riemannian Manifold

AbstractFor foliations on Riemannian manifolds, we develop elementary geometric and topological properties of the mean curvature one-form κ and the normal plane field one-form β. Through examples, we show that an important result of Kamber-Tondeur on κ is in general a best possible result. But we demonstrate that their bundle-like hypothesis can be relaxed somewhat in codimension 2. We study the structure of umbilic foliations in this more general context and in our final section establish some analogous results for flows.

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