# THE HOPF-RINOW THEOREM IN INFINITE DIMENSION

Original Publication Date: 1976-Dec-31

Included in the Prior Art Database: 2005-Sep-16

## Publishing Venue

Software Patent Institute

## Related People

Ivar Ekeland: AUTHOR [+3]

## Abstract

In this paper, I prove the following theorem. Let M be a complete, possibly infinite -dimensional, Riemannian manifold. Take any point q in M. Then the set of points p e M which can be Joined to q by a unique minimal geodesic contains a countable inter-section of open dense subsets.

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__Page 1 of 24__THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

**THE HOPF-RINOW THEOREM IN INFINITE DIMENSION **

Ivar Ekeland

Technical Summary Report # 1692

October 1976

**ABSTRACT **

In this paper, I prove the following theorem. Let M be a complete, possibly infinite -dimensional, Riemannian manifold. Take any point q in M. Then the set of points p e M which can be Joined to q by a unique minimal geodesic contains a countable inter-section of open dense subsets.

AMS(MOS) Subject Classification - 58B20Y 49AZ51 53CZO

Key Words - Hopf-Rinow theorem ' infinite -dimensional Riemannian manifolds, minimal geodesics, cutlocus Work Unit Number 1 - Applied Analysis

Sponsored by the United States Army under Contract No. DAAG29-75-C-0024,

**I. Statement of results **

We begin by reviewing some essential features. By a Riemannian manifold M we understand a connected

(Equation Omitted)

-manifold modelled on some Hilbert space H, such that the tangent space

(Equation Omitted)

carries a

00 scalar product p which is C in p e M and defines on TM p a norm p equivalent to the original norm of H

If p and q are two points in M, a path from p to q is a continuous map

(Equation Omitted)

M such that

(Equation Omitted)

The

University of Wisconsin Page 1 Dec 31, 1976

__Page 2 of 24__THE HOPF-RINOW THEOREM IN INFINITE DIMENSION

set of all piecewise C 00 paths from p to q will be denoted by C q p q q If c e C p is such a path, its length L p (c) is the real number defined

by:

(Equation Omitted)

The geodesic distance d on M is defined by:

(Equation Omitted)

It is compatible with the manifold topology of M Any path c e C q p such that

(Equation Omitted)

and the speed 11,~ is constant will be p c

00 called a minimal geodesic; it must be C . and satisfy the equation (where V denotes the Levi- Civita connection):

(Equation Omitted)

which means that b(t) is obtained from b(O) e TM p by parallel trans-

lation along c . Conversely, any solution c of (1. 3) is called a

Sponsored by the United States Army under Contract No. DAAG29-15-C-

0024. geodesic. The manifold M will often be assumed to be complete for the metric d; this will imply that solutions of (1. 3) are defined for all t E ]R~ i. e. that geodesics can be indefinitely extended.

Throughout this paper, for 5 > 0 and p e M, we shall use the following notations:

(Equation Omitted)

Whenever the solution of (1. 3) with the initial condition 6(0) E TM p

exists up to t = 1. we set

(Equation Omitted)

and call exp p the exponential map. If the Riemannian manifold M is complete, exp p ~ is
defined for all ~ e TM p . Even if it is not, by the usual theorems on differential equations (e. q.

[5], Th. IV. 1), there is a neighbourhood 'U of (0, p)

00 in TM such that the map

(Equation Omitted)

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__Page 3 of 24__THE HOPF-RINOW THEOREM IN INFINITE DIMENSION

is well-defined and C on m

Now consider the map

(Equation Omitted)

fromU to MXM. Its tangent map at (0, p) is easily seen to be an isomorphism, so...