On the Singularities of the Exponential Map in Infinite Dimensional Riemannian Manifolds

In this chapter the metric space structure of shape is developed by studying the action of the infinite dimensional diffeomorphisms on the coordinate systems of shape. Riemannian manifolds allow us to developmetric distances between the groupelements. We examine the natural analog of the finite dimensional matrix groups corresponding to the infinite dimensional diffeomorphisms which are generated as flows of ordinary differential equations.We explore the construction of the metric structure of these diffeomorphisms and develop many of the properties which hold for the finite dimensional matrix groups in this infinite dimensional setting.

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A Second Order Smooth Variational Principle on Riemannian Manifolds

Canadian Journal of Mathematics ◽

10.4153/cjm-2010-013-4 ◽

2010 ◽

Vol 62 (2) ◽

pp. 241-260 ◽

Cited By ~ 2

Author(s):

Daniel Azagra ◽

Robb Fry

Keyword(s):

Variational Principle ◽

Sectional Curvature ◽

Riemannian Manifolds ◽

Second Order ◽

Injectivity Radius ◽

Infinite Dimensional ◽

Locally Convex

AbstractWe establish a second order smooth variational principle valid for functions defined on (possibly infinite-dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.

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Metrics Spaces for the Matrix Groups

Pattern Theory ◽

10.1093/oso/9780198505709.003.0011 ◽

2006 ◽

Author(s):

Ulf Grenander ◽

Michael I. Miller

Keyword(s):

Riemannian Manifolds ◽

Metric Space ◽

Space Structure ◽

Coordinate Systems ◽

Matrix Groups ◽

Infinite Dimensional ◽

Dimensional Matrix ◽

Finite Dimensional ◽

The Matrix ◽

Metric Distances

In this chapter the metric space structure of shape is developed. We do this by first studying the action of the matrix groups on the coordinate systems of shape. We begin by reviewing the well-known properties of the finite-dimensional matrix groups, including their properties as smooth Riemannian manifolds, allowing us to develop metric distances between the groupelements. We explore the construction of the metric structure of these diffeomorphisms and develop many of the properties which hold for the finite dimensional matrix groups and subsequently in the infinite dimensional setting as well.

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Hamilton-Jacobi equations on infinite-dimensional Riemannian manifolds

Nonlinear Analysis ◽

10.1016/0362-546x(85)90015-x ◽

1985 ◽

Vol 9 (7) ◽

pp. 739-761

Author(s):

Dumitru Motreanu ◽

Cătălin Popa

Keyword(s):

Riemannian Manifolds ◽

Infinite Dimensional ◽

Jacobi Equations

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Superspace and global stability in general relativity

International Journal of Modern Physics D ◽

10.1142/s021827181741019x ◽

2017 ◽

Vol 26 (05) ◽

pp. 1741019 ◽

Cited By ~ 1

Author(s):

A. V. Gurzadyan ◽

A. A. Kocharyan

Keyword(s):

General Relativity ◽

Stability Analysis ◽

Riemannian Manifolds ◽

Jacobi Equation ◽

Global Analysis ◽

Cosmological Models ◽

Dynamical Analysis ◽

Infinite Dimensional ◽

The Stability ◽

The Universe

A framework is developed enabling the global analysis of the stability of cosmological models using the local geometric characteristics of the infinite-dimensional superspace, i.e. using the generalized Jacobi equation reformulated for pseudo-Riemannian manifolds. We give a direct formalism for dynamical analysis in the superspace, the requisite equation pertinent for stability analysis of the universe by means of generalized covariant and Fermi derivative is derived. Then, the relevant definitions and formulae are retrieved for cosmological models with a scalar field.

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On some types of geodesics on Riemannian manifolds

Nagoya Mathematical Journal ◽

10.1017/s0027763000019139 ◽

1981 ◽

Vol 81 ◽

pp. 27-43 ◽

Cited By ~ 2

Author(s):

Tetsunori Kurogi

Keyword(s):

Riemannian Manifold ◽

Critical Point ◽

Riemannian Manifolds ◽

Critical Point Theory ◽

Point Theory ◽

The Other ◽

Existence Theory ◽

Infinite Dimensional ◽

Quantitative Theory

For a given Riemannian manifold M and its submanifold N, one can find various types of geodesies on M starting from any point of N and ending in any point of N. For example, geodesies which start perpendicularly from N and end perpendicularly in N are treated by many mathematicians. K. Grove has stated a condition in a general case for the existence of such a geodesic ([4]), where he has used the method of the infinite dimensional critical point theory. This method is very useful for the study of geodesies and many geometricians have used it successfully. It has two aspects: one is an existence theory and the other is a quantitative theory, which one can find, for instance, in the excellent theory for closed geodesies of W. Klingenberg ([1], [7]) and so on.

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A free energy Lagrangian variational formulation of the Navier–Stokes–Fourier system

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887819400061 ◽

2019 ◽

Vol 16 (supp01) ◽

pp. 1940006 ◽

Cited By ~ 2

Author(s):

François Gay-Balmaz ◽

Hiroaki Yoshimura

Keyword(s):

Free Energy ◽

Riemannian Manifolds ◽

Variational Formulation ◽

Nonequilibrium Thermodynamics ◽

Discrete Systems ◽

Navier Stokes ◽

Infinite Dimensional ◽

Variational Framework ◽

Independent Variable ◽

Continuum Systems

We present a variational formulation for the Navier–Stokes–Fourier system based on a free energy Lagrangian. This formulation is a systematic infinite-dimensional extension of the variational approach to the thermodynamics of discrete systems using the free energy, which complements the Lagrangian variational formulation using the internal energy developed in [F. Gay-Balmaz and H. Yoshimura, A Lagrangian variational formulation for nonequilibrium thermodynamics, Part II: Continuum systems, J. Geom. Phys. 111 (2017) 194–212] as one employs temperature, rather than entropy, as an independent variable. The variational derivation is first expressed in the material (or Lagrangian) representation, from which the spatial (or Eulerian) representation is deduced. The variational framework is intrinsically written in a differential-geometric form that allows the treatment of the Navier–Stokes–Fourier system on Riemannian manifolds.

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L2 Extension for jets of holomorphic sections of a Hermitian line Bundle

Nagoya Mathematical Journal ◽

10.1017/s0027763000009168 ◽

2005 ◽

Vol 180 ◽

pp. 1-34 ◽

Cited By ~ 6

Author(s):

Dan Popovici

Keyword(s):

Line Bundle ◽

Riemannian Manifolds ◽

Technical Difficulty ◽

Holomorphic Section ◽

Exponential Map ◽

Extension Theorems ◽

Holomorphic Sections ◽

Final Estimate ◽

Arbitrary Compact Subset ◽

Complete Riemannian Manifolds

AbstractLet (X, ω) be a weakly pseudoconvex Kähler manifold, Y ⊂ X a closed submanifold defined by some holomorphic section of a vector bundle over X, and L a Hermitian line bundle satisfying certain positivity conditions. We prove that for any integer k > 0, any section of the jet sheaf which satisfies a certain L2 condition, can be extended into a global holomorphic section of L over X whose L2 growth on an arbitrary compact subset of X is under control. In particular, if Y is merely a point, this gives the existence of a global holomorphic function with an L2 norm under control and with prescribed values for all its derivatives up to order k at that point. This result generalizes the L2 extension theorems of Ohsawa-Takegoshi and of Manivel to the case of jets of sections of a line bundle. A technical difficulty is to achieve uniformity in the constant appearing in the final estimate. To this end, we make use of the exponential map and of a Rauch-type comparison theorem for complete Riemannian manifolds.

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