Geometry
LOCAL AND GLOBAL PROBLEMS
Problems in differential geometry can generally be divided into
two types: local problems and global problems. The tangent space
at a point and different concepts of curvature are local problems.
But the validity of a certain local property throughout the
manifold could impose strong restrictions on the manifold as
a whole; the determination of such restrictions is a global
problem. Thus, there are pieces of surfaces in Euclidean space
with constant Gaussian curvature, but the spheres are the only
surfaces that have constant Gaussian curvature and that are
closed. Also, it is a local property for a curve on a Riemannian
manifold to be a
geodesic, but the
index of a geodesic (i.e.,
the number of essentially different deformations that shorten
the geodesic with the end-points fixed) is a global invariant.
Other problems on geodesics concern the existence or nonexistence
of closed geodesics and the ergodicity of geodesic flows, both
of which are global problems.
(S.S.C.)
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