|
Showing 1 - 3 of
3 matches in All Departments
This English edition could serve as a text for a first year
graduate course on differential geometry, as did for a long time
the Chicago Notes of Chern mentioned in the Preface to the German
Edition. Suitable references for ordin ary differential equations
are Hurewicz, W. Lectures on ordinary differential equations. MIT
Press, Cambridge, Mass., 1958, and for the topology of surfaces:
Massey, Algebraic Topology, Springer-Verlag, New York, 1977. Upon
David Hoffman fell the difficult task of transforming the tightly
constructed German text into one which would mesh well with the
more relaxed format of the Graduate Texts in Mathematics series.
There are some e1aborations and several new figures have been
added. I trust that the merits of the German edition have survived
whereas at the same time the efforts of David helped to elucidate
the general conception of the Course where we tried to put Geometry
before Formalism without giving up mathematical rigour. 1 wish to
thank David for his work and his enthusiasm during the whole period
of our collaboration. At the same time I would like to commend the
editors of Springer-Verlag for their patience and good advice. Bonn
Wilhelm Klingenberg June,1977 vii From the Preface to the German
Edition This book has its origins in a one-semester course in
differential geometry which 1 have given many times at Gottingen,
Mainz, and Bonn."
The question of existence of c10sed geodesics on a Riemannian
manifold and the properties of the corresponding periodic orbits in
the geodesic flow has been the object of intensive investigations
since the beginning of global differential geo metry during the
last century. The simplest case occurs for c10sed surfaces of
negative curvature. Here, the fundamental group is very large and,
as shown by Hadamard [Had] in 1898, every non-null homotopic c10sed
curve can be deformed into a c10sed curve having minimallength in
its free homotopy c1ass. This minimal curve is, up to the
parameterization, uniquely determined and represents a c10sed
geodesic. The question of existence of a c10sed geodesic on a
simply connected c10sed surface is much more difficult. As pointed
out by Poincare [po 1] in 1905, this problem has much in common
with the problem ofthe existence of periodic orbits in the
restricted three body problem. Poincare [l.c.] outlined a proof
that on an analytic convex surface which does not differ too much
from the standard sphere there always exists at least one c10sed
geodesic of elliptic type, i. e., the corres ponding periodic orbit
in the geodesic flow is infinitesimally stable.
|
You may like...
Higher
Michael Buble
CD
(1)
R165
R138
Discovery Miles 1 380
|