Hypersurfaces with special quadric representations

1997 ◽  
Vol 56 (2) ◽  
pp. 227-234 ◽  
Author(s):  
Lu Jitan
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Isometric Immersion ◽  
Dimensional Euclidean Space ◽  
Laplacian Operator

Let x: Mn → Em be an isometric immersion of an n-dimensional Riemannian manifold into the m-dimensional Euclidean space. Then the map (where t denotes transpose) is called the quadric representation of Mn. In this paper, we study and classify hypersurfaces in the Euclidean space Em which satisfy , where B and C are two constant matrices, and Δ is the Laplacian operator of Mn. Some classification results are obtained.

scholarly journals Generalized Minkowski formulae for compact submanifolds of a Riemannian manifold

1984 ◽  
Vol 36 (3) ◽  
pp. 378-388 ◽  
Author(s):  
Geoffrey Howard Smith
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Convex Surfaces ◽  
Integral Formulae

AbstractIn 1903 H. Minkowski obtained two integral formulae for closed convex surfaces in three dimensional Euclidean space. In this paper we obtain generalised Minkowski formulae on compact orientable immersed submanifolds of an arbitrary Riemannian manifold. By successive specialisation we indicate how known integral theorems can be obtained as particular cases of our result.

Immersions with Semi-Definite Second Fundamental Forms

1975 ◽  
Vol 27 (3) ◽  
pp. 610-617 ◽  
Author(s):  
Leo B. Jonker
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Fundamental Form ◽  
Isometric Immersion ◽  
Tangent Space ◽  
Normal Component ◽  
Second Fundamental Form ◽  
Normal Space ◽  
The Second Fundamental Form ◽  
Image Position

Let M be a. complete connected Riemannian manifold of dimension n and let £:M → Rn+k be an isometric immersion into the Euclidean space Rn+k. Let ∇ be the connection on Mn and let be the Euclidean connection on Rn+k. Also letdenote the second fundamental form B(X, Y) = (xY)→. Here TP(M) denotes the tangent space at p, NP(M) the normal space and (…)→ the normal component.

scholarly journals Holonomy Groups Of Hypersurfaces

1956 ◽  
Vol 10 ◽  
pp. 8-14 ◽  
Author(s):  
Shoshichi Kobayashi
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Orthogonal Group ◽  
Closed Subgroup ◽  
Compact Riemannian Manifold ◽  
Holonomy Group ◽  
Dimensional Euclidean Space ◽  
Sufficient Condition ◽  
Compact Hypersurface ◽  
Proper Orthogonal

The restricted homogeneous holonomy group of an n–dimensional Riemannian manifold is a connected closed subgroup of the proper orthogonal group SO(n) [1]. In this note we shall prove that the restricted homogeneous holonomy group of an n-dimensional compact hypersurface in the Euclidean space is actually the proper orthogonal group SO(n) itself. This gives a necessary (of course, not sufficient) condition for the imbedding of an n-dimensional compact Riemannian manifold into the (n +1)–dimensional Euclidean space.

scholarly journals Isometric immersion of a compact Riemannian manifold into a Euclidean space

1992 ◽  
Vol 46 (2) ◽  
pp. 177-178 ◽  
Author(s):  
Sharief Deshmukh
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Scalar Curvature ◽  
Ricci Curvature ◽  
Isometric Immersion ◽  
Compact Riemannian Manifold

We show that an isometric immersion of an n−dimensional compact Riemannian manifold of non-negative Ricci curvature with scalar curvature always less than n(n−1)λ−2 into a Euclidean space of dimension n + 1 can never be contained in a ball of radius λ.

scholarly journals Integral Formulas for Submanifolds and their Applications

1970 ◽  
Vol 22 (2) ◽  
pp. 376-388 ◽  
Author(s):  
Kentaro Yano
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Mean Curvature ◽  
Constant Mean Curvature ◽  
Dimensional Euclidean Space ◽  
3 Dimensional ◽  
Integral Formulas ◽  
Closed Hypersurfaces

Liebmann [12] proved that the only ovaloids with constant mean curvature in a 3-dimensional Euclidean space are spheres. This result has been generalized to the case of convex closed hypersurfaces in an m-dimensional Euclidean space by Alexandrov [1], Bonnesen and Fenchel [3], Hopf [4], Hsiung [5], and Süss [14].The result has been further generalized to the case of closed hypersurfaces in an m-dimensional Riemannian manifold by Alexandrov [2], Hsiung [6], Katsurada [7; 8; 9], Ōtsuki [13], and by myself [15; 16].The attempt to generalize the result to the case of closed submanifolds in an m-dimensional Riemannian manifold has been recently done by Katsurada [10; 11], Kôjyô [10], and Nagai [11].

On Vitali's Theorem for Groups of Motions of Euclidean Space

1999 ◽  
Vol 6 (4) ◽  
pp. 323-334
Author(s):  
A. Kharazishvili
Keyword(s):  
Euclidean Space ◽  
Dimensional Euclidean Space ◽  
Finite Dimensional

Abstract We give a characterization of all those groups of isometric transformations of a finite-dimensional Euclidean space, for which an analogue of the classical Vitali theorem [Sul problema della misura dei gruppi di punti di una retta, 1905] holds true. This characterization is formulated in purely geometrical terms.

Some properties of Wigner coefficients and hyperspherical harmonics

1956 ◽  
Vol 52 (3) ◽  
pp. 424-430 ◽  
Author(s):  
A. P. Stone
Keyword(s):  
Angular Momentum ◽  
Euclidean Space ◽  
Rotation Group ◽  
Dimensional Euclidean Space ◽  
Hyperspherical Harmonics ◽  
Shift Operators ◽  
Analytical Relations

ABSTRACTGeneral shift operators for angular momentum are obtained and applied to find closed expressions for some Wigner coefficients occurring in a transformation between two equivalent representations of the four-dimensional rotation group. The transformation gives rise to analytical relations between hyperspherical harmonics in a four-dimensional Euclidean space.

On inequalities of the Tchebychev type

1963 ◽  
Vol 59 (1) ◽  
pp. 135-146 ◽  
Author(s):  
J. F. C. Kingman
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Random Variable ◽  
Dimensional Euclidean Space ◽  
Real Random Variable ◽  
Finite Dimensional

1. A type of problem which frequently occurs in probability theory and statistics can be formulated in the following way. We are given real-valued functions f(x), gi(x) (i = 1, 2, …, k) on a space (typically finite-dimensional Euclidean space). Then the problem is to set bounds for Ef(X), where X is a random variable taking values in , about which all we know is the values of Egi(X). For example, we might wish to set bounds for P(X > a), where X is a real random variable with some of its moments given.

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