Contemporary Mathematics

Volume 359, 2004

A Beemian Sampler: 1966-2002

Paul E. Ehrlich and Kevin L. Easley

ABSTRACT.

A survey is given of aspects of the development of global space-time differ-

ential geometry since the 1970s, focusing on some major areas of investigation

of Professor John Beem, various co-authors, and Ph.D. students.

1.

Introduction

In accordance with the overall theme of the Beemfest, Advances in Differ-

ential Geometry and General Relativity, a selective summary is presented of the

research of Professor John Beem and various co-authors and students, beginning

with Beem's first joint publication in 1966. Since a time period of nearly four

decades is covered, results discussed are placed in historical context. As an aspect

of this viewpoint, the language employed in the original papers is often used.

Any student of Riemannian geometry is exposed to a wonderful global result

and basic working tool, which goes back to Hopf and Rinow [65].

If (N,g0

)

is a

Riemannian manifold, then an associated Riemannian distance function

do : N

x

N

---+

JR. is given by

do(p, q)

=

inf{L(c)

I

c: [0, 1]

---+

N

is a piecewise

smooth curve with c(O)

=

p,

c(1)

=

q}.

Then the promised result guarantees the equivalence of the following conditions:

HoPF-RINOW THEOREM.

For any Riemannian manifold (N,go), the following

are equivalent:

1.

metric completeness:

(N, d0

)

is a complete metric space;

2. geodesic completeness:

for any v

E

TN, the geodesic cv(t) in N with

initial condition cv'(O)

=

v is defined for all values of an affine parameter t;

3.

for some point p

E

N, the exponential map expp is defined on all of TpN;

4. finite compactness:

every subset K of N that is do-bounded has compact

closure.

1991 Mathematics Subject Classification. Primary 53C50, 01A65.

©

2004 American Mathematical Society

http://dx.doi.org/10.1090/conm/359/06552