Page 143 - Curvature and Homology

EXERCISES

Consequently,
au = K(~) &,, (i: not summed)

from which we derive

By employing theorem 3.2.4 show that if M is compact and orientable, then
bl(M) = bdM) = 0.
2. If at each point of M, the ratio of the largest to the smallest principal curvature
is at most a, M is a homology sphere.
Hint: Apply theorem 3.2.6.
3. If M is locally isometrically imbedded in an (n + 1)-dimensional space of
positive constant curvature K, the Gauss equations are given by

Show that the assertions in A.l and A.2 are also valid in this case.
4. If the mean curoatwe of the hypersurface vanishes, that is, if, in terms of the
metric g of M,
gZj a, = 0
then, from the representation of the curvature tensor given in A.1

where

In this case, M is called a minimal hypersurface or a minimal variety of @+I,
Show that the only groups of motions of a compact and orientable minimal
variety are groups of translations.
5. Show that the only groups of motions of a compact and orientable minimal
variety (hypersurface of zero mean curvature) imbedded in a manifold of
constant negative curvature are translation groups.
6. If all the geodesics of a hypersurface M are also geodesics of the space in
which it is imbedded, M is called a totally gcodwic hyperwface. It is
known that a totally geodesic hypersurface is a minimal variety. Hence, if it is
compact and orientable and, if the imbedding space is a manifold of constant
non-positive curvature its only groups of motions are translation groups.