On the global isometric embedding of pseudo-Riemannian manifolds

1970 ◽  
Vol 314 (1518) ◽  
pp. 417-428 ◽  
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
General Result ◽  
Isometric Embedding ◽  
Positive Definite ◽  
Space Time ◽  
Lower Dimension ◽  
Definite Case

It is shown that any pseudo-Riemannian manifold has (in Nash’s sense) a proper isometric embedding into a pseudo-Euclidean space, which can be made to be of arbitrarily high differentiability. The application of this to the positive definite case treated by Nash gives a new proof using a Euclidean space of substantially lower dimension. The general result is applied to the space-time of relativity, and the dimensions and signatures of the spaces needed to embed various cases are evaluated.

A Note on Hyperconvexity in Riemannian Manifolds

1959 ◽  
Vol 11 ◽  
pp. 576-582
Author(s):  
Albert Nijenhuis
Keyword(s):  
Riemannian Manifold ◽  
Curvature Tensor ◽  
Shortest Path ◽  
Riemannian Manifolds ◽  
Convex Set ◽  
Geodesic Segment ◽  
Positive Definite ◽  
Classical Theorem ◽  
Arc Length ◽  
Class C

Let M denote a connected Riemannian manifold of class C3, with positive definite C2 metric. The curvature tensor then exists, and is continuous.By a classical theorem of J. H. C. Whitehead (1), every point x of M has the property that all sufficiently small spherical neighbourhoods V of x are convex; that is, (i) to every y,z ∈ V there is one and only one geodesic segment yz in M which is the shortest path joining them:f:([0, 1]) → M,f(0) = y, f(1) = z; and (ii) this segment yz lies entirely in V:f([0, 1]) V; (iii) if f is parametrized proportional to arc length, then f(t) is a C2 function of y, t, and z.Let V be a convex set in M; and let y1 y2, Z1, z2 ∈ V.

scholarly journals On the Differential Equation Governing Torqued Vector Fields on a Riemannian Manifold

Symmetry ◽  
2020 ◽  
Vol 12 (12) ◽  
pp. 1941
Author(s):  
Sharief Deshmukh ◽  
Nasser Bin Turki ◽  
Haila Alodan
Keyword(s):  
Riemannian Manifold ◽  
Vector Field ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Vector Fields ◽  
Compact Riemannian Manifold ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Rigidity Results ◽  
Necessary And Sufficient

In this article, we show that the presence of a torqued vector field on a Riemannian manifold can be used to obtain rigidity results for Riemannian manifolds of constant curvature. More precisely, we show that there is no torqued vector field on n-sphere Sn(c). A nontrivial example of torqued vector field is constructed on an open subset of the Euclidean space En whose torqued function and torqued form are nowhere zero. It is shown that owing to topology of the Euclidean space En, this type of torqued vector fields could not be extended globally to En. Finally, we find a necessary and sufficient condition for a torqued vector field on a compact Riemannian manifold to be a concircular vector field.

scholarly journals Isometric Embeddings of Pro-Euclidean Spaces

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Barry Minemyer
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Metric Space ◽  
Inverse Limit ◽  
Isometric Embedding ◽  
Embedding Theorem ◽  
Euclidean Spaces ◽  
Compact Metric Space ◽  
Covering Dimension ◽  
Isometric Embeddings

Abstract In [12] Petrunin proves that a compact metric space X admits an intrinsic isometry into En if and only if X is a pro-Euclidean space of rank at most n, meaning that X can be written as a “nice” inverse limit of polyhedra. He also shows that either case implies that X has covering dimension at most n. The purpose of this paper is to extend these results to include both embeddings and spaces which are proper instead of compact. The main result of this paper is that any pro-Euclidean space of rank at most n is proper and admits an intrinsic isometric embedding into E2n+1. Since every n-dimensional Riemannian manifold is a pro-Euclidean space of rank at most n, this result is a partial generalization of (the C0 version of) the famous Nash isometric embedding theorem from [10].

scholarly journals MATHEMATICAL SUPPORT TO BRANEWORLD THEORY

2007 ◽  
Vol 04 (08) ◽  
pp. 1259-1267 ◽  
Author(s):  
EDMUNDO M. MONTE
Keyword(s):  
Euclidean Space ◽  
High Dimension ◽  
Riemannian Manifolds ◽  
Fundamental Theorem ◽  
New Physics ◽  
Space Time ◽  
Bulk Solution ◽  
Fundamental Interactions ◽  
Integrability Conditions ◽  
The Law

The braneworld theory appear with the purpose of solving the problem of the hierarchy of the fundamental interactions. The perspectives of the theory emerge as a new physics, for example, deviation of the law of Newton's gravity. One of the principles of the theory is to suppose that the braneworld is local submanifold in a space of high dimension, the bulk, solution of Einstein's equations in high dimension. In this paper we approach the mathematical consistency of this theory with a new proof of the fundamental theorem of submanifolds for the case of semi-Riemannian manifolds. This theorem consists of an essential mathematical support for this new theory. We find the integrability conditions for the existence of space–time submanifolds in a pseudo-Euclidean space.

Isometry of Riemannian Manifolds to Spheres, II

1976 ◽  
Vol 28 (1) ◽  
pp. 63-72 ◽  
Author(s):  
Neill H. Ackerman ◽  
C. C. Hsiung
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Symmetric Matrix ◽  
Riemann Tensor ◽  
Positive Definite ◽  
Inverse Matrix ◽  
Covariant Differentiation

Let Mn be a Riemannian manifold of dimension n ≧ 2 and class C3, (gtj) the symmetric matrix of the positive definite metric of Mn, and (gij) the inverse matrix of (gtj), and denote by and R = gijRij the operator of covariant differentiation with respect to gij, the Riemann tensor, the Ricci tensor and the scalar curvature of Mn respectively.

scholarly journals A Class of Riemannian Manifolds Satisfying R(X ,Y)·R = 0

1971 ◽  
Vol 42 ◽  
pp. 67-77 ◽  
Author(s):  
Shûkichi Tanno
Keyword(s):  
Riemannian Manifold ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Additional Condition ◽  
Affirmative Answer ◽  
Tensor Algebra ◽  
Riemannian Curvature ◽  
Algebraic Condition ◽  
Locally Symmetric Space ◽  
Riemannian Curvature Tensor

Let (M,g) be a Riemannian manifold and let R be its Riemannian curvature tensor. If (M, g) is a locally symmetric space, we have(*) R(X,Y)·R = 0 for all tangent vectors X,Ywhere the endomorphism R(X,Y) (i.e., the curvature transformation) operates on R as a derivation of the tensor algebra at each point of M. There is a question: Under what additional condition does this algebraic condition (*) on R imply that (M,g) is locally symmetric (i.e., ∇R = 0)? A conjecture by K. Nomizu [5] is as follows : (*) implies ∇R = 0 in the case where (M, g) is complete and irreducible, and dim M ≥ 3. He gave an affirmative answer in the case where (M,g) is a certain complete hypersurface in a Euclidean space ([5]).

scholarly journals On Compact Riemannian Manifolds with Zero Ricci Curvature

1956 ◽  
Vol 10 (3) ◽  
pp. 131-133 ◽  
Author(s):  
T. J. Willmore
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Betti Number ◽  
Positive Definite ◽  
Arbitrary Signature ◽  
Local Result

In this note I prove the following result:—Theorem. A compact, orientable, Riemannian manifold Mn, with positive definite metric and zero Ricci curvature, is flat if the first Betti number R1 exceeds n — 4.In this statement of the theorem it is assumed that the dimensions of Mn are not less than four. If this is not the case, the result is still valid but appears as a purely local result and is true for a metric of arbitrary signature.

Mean-value theorems for Riemannian manifolds

1982 ◽  
Vol 92 (3-4) ◽  
pp. 343-364 ◽  
Author(s):  
A. Gray ◽  
T. J. Willmore
Keyword(s):  
Riemannian Manifold ◽  
Harmonic Function ◽  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Integrable Function ◽  
Mean Value ◽  
Geodesic Sphere ◽  
Einstein Spaces ◽  
The Mean ◽  
Image Position

SynopsisLet Mm (r, f) denote the mean-value of a real-valued integrable function f over a geodesic sphere with centre m and radius r in an n-dimensional Riemannian manifold M. We obtain an expansion of Mm (r, f) in powers of r, thereby generalizing Pizzetti's formula valid in euclidean space. From this expansion we prove that the propertyfor every harmonic function near m, characterizes Einstein spaces. We define super-Einstein spaces and prove that they are characterized by the property

scholarly journals $L^{p}$ harmonic 1-forms on conformally flat Riemannian manifolds

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jing Li ◽  
Shuxiang Feng ◽  
Peibiao Zhao
Keyword(s):  
Riemannian Manifold ◽  
Riemannian Manifolds ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Finiteness Theorem ◽  
Conformally Flat ◽  
Locally Conformally Flat ◽  
Flat Riemannian Manifold ◽  
Ricci Form

AbstractIn this paper, we establish a finiteness theorem for $L^{p}$ L p harmonic 1-forms on a locally conformally flat Riemannian manifold under the assumptions on the Schrödinger operators involving the squared norm of the traceless Ricci form. This result can be regarded as a generalization of Han’s result on $L^{2}$ L 2 harmonic 1-forms.

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