scholarly journals An integral formula for hypersurfaces in space forms

1995 ◽  
Vol 37 (3) ◽  
pp. 337-341 ◽  
Author(s):  
Theodoros Vlachos
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Distance Function ◽  
Integral Formula ◽  
Constant Sectional Curvature ◽  
Position Vector ◽  
Space Forms ◽  
Image Position ◽  
Simply Connected

Let be an n+ 1-dimensional, complete simply connected Riemannian manifold of constant sectional curvature c and We consider the function r(·) = d(·, P0) where d stands for the distance function in and we denote by grad r the gradient of The position vector (see [1]) with origin P0 is defined as where ϕ(r)equalsr, if c = 0, c< 0 or c <0 respectively.

scholarly journals Some results on stable p-harmonic maps

1994 ◽  
Vol 36 (1) ◽  
pp. 77-80 ◽  
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Riemannian Manifold ◽  
Critical Point ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Complete Riemannian Manifold ◽  
Image Position ◽  
Simply Connected

For each p ∈ [2, ∞)a p-harmonic map f:Mm→Nn is a critical point of the p-energy functionalwhere Mm is a compact and Nn a complete Riemannian manifold of dimensions m and n respectively. In a recent paper [3], Takeuchi has proved that for a certain class of simply-connected δ-pinched Nn and certain type of hypersurface Nn in ℝn+1, the only stable p-harmonic maps for any compact Mm are the constant maps. Our purpose in this note is to establish the following theorem which complements Takeuchi's results.

scholarly journals The double cover relative to a convex domain and the relative isoperimetric inequality

2006 ◽  
Vol 80 (3) ◽  
pp. 375-382 ◽  
Author(s):  
Jaigyoung Choe
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Isoperimetric Inequality ◽  
Convex Domain ◽  
Double Cover ◽  
Nonpositive Sectional Curvature ◽  
Relative Isoperimetric Inequality ◽  
Simply Connected

AbstractWe prove that a domain Ω in the exterior of a convex domain C in a four-dimensional simply connected Riemannian manifold of nonpositive sectional curvature satisfies the relative isoperimetric inequality 64π2 Vol(Ω)3 < Vol(∂Ω ~ ∂C)4. Equality holds if and only if Ω is an Euclidean half ball and ∂Ω ~ ∂C is a hemisphere.

scholarly journals A note on complete hypersurfaces of non-positive Ricci curvature

1983 ◽  
Vol 28 (3) ◽  
pp. 339-342 ◽  
Author(s):  
G.H. Smith
Keyword(s):  
Euclidean Space ◽  
Recent Result ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Constant Sectional Curvature ◽  
Positive Ricci Curvature ◽  
Simply Connected ◽  
Complete Hypersurfaces

In this note we point out that a recent result of Leung concerning hypersurfaces of a Euclidean space has a simple generalisation to hypersurfaces of complete simply-connected Riemannian manifolds of non-positive constant sectional curvature.

Curvature of K-contact Semi-Riemannian Manifolds

2014 ◽  
Vol 57 (2) ◽  
pp. 401-412 ◽  
Author(s):  
Domenico Perrone
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Sectional Curvature ◽  
Riemannian Manifolds ◽  
Lorentzian Manifold ◽  
Constant Sectional Curvature ◽  
Conformally Flat ◽  
Reeb Vector ◽  
Reeb Vector Field

Abstract.In this paper we characterize K-contact semi-Riemannian manifolds and Sasakian semi- Riemannian manifolds in terms of curvature. Moreover, we show that any conformally flat K-contact semi-Riemannian manifold is Sasakian and of constant sectional curvature κ = ɛ, where ɛ = ± denotes the causal character of the Reeb vector field. Finally, we give some results about the curvature of a K-contact Lorentzian manifold.

scholarly journals Indefinite Almost Paracontact Metric Manifolds

2010 ◽  
Vol 2010 ◽  
pp. 1-19 ◽  
Author(s):  
Mukut Mani Tripathi ◽  
Erol Kılıç ◽  
Selcen Yüksel Perktaş ◽  
Sadık Keleş
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Curvature Tensor ◽  
Ricci Tensor ◽  
Sasakian Manifold ◽  
Constant Sectional Curvature ◽  
Sasakian Manifolds ◽  
Sasakian Structure

We introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para-Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it cannot admit an (ε)-para Sasakian structure. We show that, for an (ε)-para Sasakian manifold, the conditions of being symmetric, semi-symmetric, or of constant sectional curvature are all identical. It is shown that a symmetric spacelike (resp., timelike) (ε)-para Sasakian manifoldMnis locally isometric to a pseudohyperbolic spaceHνn(1)(resp., pseudosphereSνn(1)). At last, it is proved that for an (ε)-para Sasakian manifold the conditions of being Ricci-semi-symmetric, Ricci-symmetric, and Einstein are all identical.

scholarly journals AN INTRINSIC PROOF OF GROMOLL–GROVE DIAMETER RIGIDITY THEOREM

2007 ◽  
Vol 09 (03) ◽  
pp. 401-419 ◽  
Author(s):  
JIANGUO CAO ◽  
HONGYAN TANG
Keyword(s):  
Riemannian Manifold ◽  
Sectional Curvature ◽  
Projective Spaces ◽  
Rigidity Theorem ◽  
Simply Connected

Using the spherical trip theorem, we present a new intrinsic proof of Gromoll–Grove diameter rigidity theorem: "If a simply-connected Riemannian manifold has sectional curvature ≥ 1 and diameter [Formula: see text], then either it is homeomorphic to a sphere, or it is isometric to one of classic projective spaces".

Biharmonic hypersurfaces in pseudo-Riemannian space forms

2016 ◽  
Vol 13 (07) ◽  
pp. 1650094 ◽  
Author(s):  
Dan Yang ◽  
Yu Fu
Keyword(s):  
Sectional Curvature ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
Constant Sectional Curvature ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

Let [Formula: see text] be a nondegenerate biharmonic pseudo-Riemannian hypersurface in a pseudo-Riemannian space form [Formula: see text] with constant sectional curvature [Formula: see text]. We show that [Formula: see text] has constant mean curvature provided that it has three distinct principal curvatures and the Weingarten operator can be diagonalizable.

Explicit construction of Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c)

2002 ◽  
Vol 132 (3) ◽  
pp. 481-508 ◽  
Author(s):  
YUN MYUNG OH
Keyword(s):  
Sectional Curvature ◽  
Isometric Immersion ◽  
Complex Space ◽  
Space Form ◽  
Real Space ◽  
Constant Sectional Curvature ◽  
Complex Space Form ◽  
Product Decomposition ◽  
Twisted Product ◽  
Space Forms

In [4], it is proved that there exists a ‘unique’ adapted Lagrangian isometric immersion of a real-space-form Mn(c) of constant sectional curvature c into a complex-space-form M˜n(4c) of constant sectional curvature 4c associated with each twisted product decomposition of a real-space-form if its twistor form is twisted closed. Conversely, if L: Mn(c) → M˜n(4c) is a non-totally geodesic Lagrangian isometric immersion of a real-space-form Mn(c) into a complex-space-form M˜n(4c), then Mn(c) admits an appropriate twisted product decomposition with twisted closed twistor form and, moreover, the immersion L is determined by the corresponding adapted Lagrangian isometric immersion of the twisted product decomposition. It is natural to ask the explicit expressions of adapted Lagrangian isometric immersions of twisted product decompositions of real-space-forms Mn(c) into complex-space-forms M˜n(4c) for each case: c = 0, c > 0 and c < 0.

Sign in / Sign up

Export Citation Format

Share Document