Union Curves of a Hypersurface

1950 ◽  
Vol 2 ◽  
pp. 457-460 ◽  
Author(s):  
C. E. Springer
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Curvature Vector ◽  
Totally Geodesic ◽  
Ordinary Space ◽  
Union Curve ◽  
Straight Lines

1. Introduction. A curve on an ordinary surface is a union curve if its osculating plane at each point contains the line of a specified rectilinear congruence through the point. The author has obtained the differential equations of union curves on a metric surface in ordinary space and has exhibited certain generalizations for union curves of known results concerning geodesic curves on a surface. It is the purpose of the present paper to develop the differential equations of the union curves of a hypersurface Vn immersed in a Riemannian manifold Vn+1 of n + 1 dimensions. The osculating plane to a curve on a surface is generalized to a totally geodesic surface the straight lines of which are geodesies in the space Vn+1. A formula is given for the union curvature vector of a curve in Vn.

scholarly journals Half lightlike submanifolds of a semi-Riemannian manifold of quasi-constant curvature

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1737-1745
Author(s):  
Dae Jin ◽  
Jae Lee
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Constant Curvature ◽  
Curvature Vector ◽  
Conformal Killing ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Screen Distribution ◽  
Lightlike Submanifolds

We study the geometry of half lightlike submanifolds (M,g,S(TM), S(TM?)) of a semi-Riemannian manifold (M~,g~) of quasi-constant curvature subject to the following conditions; (1) the curvature vector field ? of M~ is tangent to M, (2) the screen distribution S(TM) of M is either totally geodesic or totally umbilical in M, and (3) the co-screen distribution S(TM?) of M is a conformal Killing distribution.

scholarly journals Examples and classification of Riemannian submersions satisfying a basic equality

2005 ◽  
Vol 72 (3) ◽  
pp. 391-402 ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Riemannian Manifold ◽  
Unit Sphere ◽  
Isometric Immersion ◽  
Riemannian Submersion ◽  
Sharp Inequality ◽  
Riemannian Submersions ◽  
Equality Case ◽  
Totally Geodesic ◽  
Basic Equality

In an earlier article we obtain a sharp inequality for an arbitrary isometric immersion from a Riemannian manifold admitting a Riemannian submersion with totally geodesic fibres into a unit sphere. In this article we investigate the immersions which satisfy the equality case of the inequality. As a by-product, we discover a new characterisation of Cartan hypersurface in S4.

A Tensor Equation of Elliptic Type

1953 ◽  
Vol 5 ◽  
pp. 524-535 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Riemannian Manifold ◽  
Differential Equations ◽  
Tensor Field ◽  
Differential Operators ◽  
Elliptic Type ◽  
Tensor Equation ◽  
Tensor Theory ◽  
Elliptic Differential Equations ◽  
Invariant Differential ◽  
Image Position

The theory of the systems of partial differential equations which arise in connection with the invariant differential operators on a Riemannian manifold may be developed by methods based on those of potential theory. It is therefore natural to consider in the same context the theory of elliptic differential equations, in particular those which are self-adjoint. Some results for a tensor equation in which appears, in addition to the operator Δ of tensor theory, a matrix or double tensor field defined on the manifold, are here presented. The equation may be writtenin a notation explained below.

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.

SOME RESULTS ON HARMONIC MAPS FOR FINSLER MANIFOLDS

2005 ◽  
Vol 16 (09) ◽  
pp. 1017-1031 ◽  
Author(s):  
QUN HE ◽  
YI-BING SHEN
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Maps ◽  
Harmonic Map ◽  
Energy Functional ◽  
Second Variation ◽  
Finsler Manifolds ◽  
Totally Geodesic ◽  
The Stability ◽  
The Second Variation ◽  
Weitzenböck Formula

By simplifying the first and the second variation formulas of the energy functional and generalizing the Weitzenböck formula, we study the stability and the rigidity of harmonic maps between Finsler manifolds. It is proved that any nondegenerate harmonic map from a compact Einstein Riemannian manifold with nonnegative scalar curvature to a Berwald manifold with nonpositive flag curvature is totally geodesic and there is no nondegenerate stable harmonic map from a Riemannian unit sphere Sn (n > 2) to any Finsler manifold.

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 Surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space

Filomat ◽  
2019 ◽  
Vol 33 (4) ◽  
pp. 1135-1145
Author(s):  
Georgi Ganchev ◽  
Velichka Milousheva
Keyword(s):  
Vector Field ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Curvature Vector ◽  
Mean Curvature Vector ◽  
Partial Differential ◽  
Invariant Functions ◽  
Systems Of Pdes

We study surfaces with parallel normalized mean curvature vector field in Euclidean or Minkowski 4-space. On any such surface we introduce special isothermal parameters (canonical parameters) and describe these surfaces in terms of three invariant functions. We prove that any surface with parallel normalized mean curvature vector field parametrized by canonical parameters is determined uniquely up to a motion in Euclidean (or Minkowski) space by the three invariant functions satisfying a system of three partial differential equations. We find examples of surfaces with parallel normalized mean curvature vector field and solutions to the corresponding systems of PDEs in Euclidean or Minkowski space in the class of the meridian surfaces.

scholarly journals Foliation by G-orbits

2014 ◽  
Vol 33 (2) ◽  
pp. 79-87
Author(s):  
Jose Rosales-Ortega
Keyword(s):  
Riemannian Manifold ◽  
Lie Group ◽  
Normal Bundle ◽  
Semisimple Lie Group ◽  
Totally Geodesic

We study the properties of the normal bundle defined by the bundle of the G-orbits of the action of a semisimple Lie group G on a pseudo Riemannian manifold M, as a consequence we obtain that the foliation induced by the normal bundle is integrable and totally geodesic.

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