scholarly journals Isometric immersion of flat Riemannian manifolds in Euclidean space.

1962 ◽  
Vol 9 (3) ◽  
pp. 199-205 ◽  
Author(s):  
Barrett O'Neill
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Isometric Immersion

SMOLYANOV–WEIZSÄCKER SURFACE MEASURES GENERATED BY DIFFUSIONS ON THE SET OF TRAJECTORIES IN RIEMANNIAN MANIFOLDS

2008 ◽  
Vol 11 (01) ◽  
pp. 21-31 ◽  
Author(s):  
ILYA V. TELYATNIKOV
Keyword(s):  
Brownian Motion ◽  
Euclidean Space ◽  
Diffusion Process ◽  
Riemannian Manifolds ◽  
Marginal Distribution ◽  
Diffusion Processes ◽  
Ambient Space ◽  
Time Interval ◽  
Standard Brownian Motion ◽  
Limit Surface

We consider surface measures on the set of trajectories in a smooth compact Riemannian submanifold of Euclidean space generated by diffusion processes in the ambient space. A construction of surface measures on the path space of a smooth compact Riemannian submanifold of Euclidean space was introduced by Smolyanov and Weizsäcker for the case of the standard Brownian motion. The result presented in this paper extends the result of Smolyanov and Weizsäcker to the case when we consider measures generated by diffusion processes in the ambient space with nonidentical correlation operators. For every partition of the time interval, we consider the marginal distribution of the diffusion process in the ambient space under the condition that it visits the manifold at all times of the partition, when the mesh of the partition tends to zero. We prove the existence of some limit surface measures and the equivalence of the above measures to the distribution of some diffusion process on the manifold.

scholarly journals Nonexistence for hyperbolic problems on Riemannian manifolds1

2020 ◽  
Vol 120 (1-2) ◽  
pp. 87-101
Author(s):  
Dario D. Monticelli ◽  
Fabio Punzo ◽  
Marco Squassina
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Wave Operator ◽  
Existence Of Solutions ◽  
Necessary Conditions ◽  
Hyperbolic Problems ◽  
Power Nonlinearity ◽  
General Weight ◽  
Noncompact Riemannian Manifolds ◽  
General Weight Function

We establish necessary conditions for the existence of solutions to a class of semilinear hyperbolic problems defined on complete noncompact Riemannian manifolds, extending some nonexistence results for the wave operator with power nonlinearity on the whole Euclidean space. A general weight function depending on spacetime is allowed in front of the power nonlinearity.

scholarly journals A note on bounded variation and heat semigroup on Riemannian manifolds

2007 ◽  
Vol 76 (1) ◽  
pp. 155-160 ◽  
Author(s):  
A. Carbonaro ◽  
G. Mauceri
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Bounded Variation ◽  
Riemannian Manifolds ◽  
Heat Semigroup ◽  
Functions Of Bounded Variation ◽  
Fixed Radius ◽  
Lower Bound Estimate

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

SOME EXISTENCE AND NONEXISTENCE RESULTS OF ISOMETRIC IMMERSIONS OF RIEMANNIAN MANIFOLDS

2004 ◽  
Vol 06 (06) ◽  
pp. 867-879 ◽  
Author(s):  
ZIZHOU TANG
Keyword(s):  
Euclidean Space ◽  
Projective Plane ◽  
Riemannian Manifolds ◽  
Unit Sphere ◽  
Klein Bottle ◽  
Euler Class ◽  
Isometric Immersions ◽  
Existence And Nonexistence

This paper investigates existence and non-existence of immersions of Riemannian manifolds. It discovers the lowest dimension of the Euclidean space into which the projective plane FP2 is isometrically immersed, by the computation of the normal Euler class. For strictly hyperbolic immersion, a new obstruction involving signature or Kervaire semi-characteristic is found. As for the existence, it constructs a strictly hyperbolic immersion from the Klein bottle to the unit sphere S3(1), solving a question posed by Gromov.

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.

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 On Certain Mappings of Riemannian Manifolds

1965 ◽  
Vol 25 ◽  
pp. 121-142
Author(s):  
Minoru Kurita
Keyword(s):  
Differential Geometry ◽  
Euclidean Space ◽  
Conformal Mapping ◽  
Riemannian Manifolds ◽  
Stereographic Projection ◽  
Point Of View ◽  
The Other ◽  
Central Projection ◽  
Special Cases ◽  
Special Case

In this paper we consider certain tensors associated with differentiable mappings of Riemannian manifolds and apply the results to a p-mapping, which is a special case of a subprojective one in affinely connected manifolds (cf. [1], [7]). The p-mapping in Riemannian manifolds is a generalization of a conformal mapping and a projective one. From a point of view of differential geometry an analogy between these mappings is well known. On the other hand it is interesting that a stereographic projection of a sphere onto a plane is conformal, while a central projection is projectve, namely geodesic-preserving. This situation was clarified partly in [6]. A p-mapping defined in this paper gives a precise explanation of this and also affords a certain mapping in the euclidean space which includes a similar mapping and an inversion as special cases.

scholarly journals PTOLEMAIC SPACES AND CAT(0)

2009 ◽  
Vol 51 (2) ◽  
pp. 301-314 ◽  
Author(s):  
S. M. BUCKLEY ◽  
K. FALK ◽  
D. J. WRAITH
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Metric Space ◽  
Metric Spaces ◽  
Finsler Manifold ◽  
Full Generality ◽  
Space Setting ◽  
Ptolemaic Space

AbstractWe consider Ptolemy's inequality in a metric space setting. It is not hard to see that CAT(0) spaces satisfy this inequality. Although the converse is not true in full generality, we show that if our Ptolemaic space is either a Riemannian or Finsler manifold, then it must also be CAT(0). Ptolemy's inequality is closely related to inversions of metric spaces. We exploit this link to establish a new characterization of Euclidean space amongst all Riemannian manifolds.

scholarly journals A theorem of Archimedes about spheres and cylinders and two-point homogeneous spaces

1991 ◽  
Vol 43 (2) ◽  
pp. 283-294 ◽  
Author(s):  
M. Djorić ◽  
L. Vanhecke
Keyword(s):  
Euclidean Space ◽  
Riemannian Manifolds ◽  
Homogeneous Spaces ◽  
Three Dimensional ◽  
Dimensional Euclidean Space ◽  
Classical Theorem ◽  
Geodesic Spheres

Starting from the well-known classical theorem of Archimedes about the volumes of spheres and circumscribing cylinders in three-dimensional Euclidean space, one considers circumscribing tubes of small geodesic spheres in general Riemannian manifolds and one derives new characterisations of two-point homogeneous spaces from it.

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