Page 260 - Curvature and Homology

242 VI. CURVATURE AND HOMOLOGY: KAEHLER MANIFOLDS

Then,

3. Prove that the curvature tensor (defined by the invariant metric g) of a
symmetric space has vanishing covariant derivative.
Hint: Make essential use of the fact that an illvariant form on a symmetric
space is a closed form.

F. Bochner's lemma [4]

1. Let M be a differentiable manifold and U a coordinate neighborhood of M
with the local coordinates (ul, ..., un). Consider the elliptic operator

on F(U)-the algebra of differentiable functions of class 2 on U, where the
coefficients gjk, hi are merely assumed to be continuous functions on U. (The
condition that L is elliptic is equivalent to the condition that the matrix Vk)
is positive definite). If for an element f E F : (a) Lf 2 0 and (b) f(ul, *.-, un) 5
f (a1, .. ., an) for some point Po E U with coordinates (a1, . ., an), then f (ul, .., un)
= f(al, -.., an) everywhere in U.
This maximum principle is due to E. Hopf [40]. The corresponding minimum
principle is given by reversing the inequalities. This result should be compared
with (V. A. 2).
2. If M is compact and f e F(M) is a differentiable function (of class 2) for
which Lf 2 0, then f is a constant function on M.
3. If M is a compact Riemannian manifold, then a function f E F(M) for which
Af 2 0 is a constant function on M.
This is the Bochner lemma [#I.
Note that M need not be orientable. By applying the Hopf minimum principle
the statements 2 and 3 are seen to be valid with the inequalities reversed.

G. Zero curvature

1. The results of 5 6.7 may be described in the following manner:
Zero curvature is the integrability condition for the pfaffian system given
by the connection forms on the bundle B of unitary frames over M. Hence,
there exist integral manifolds; a maximal integral submanifold through a point
will be a covering space of the manifold M. These manifolds are locally isometric
i since the mapping from the horizontal part of the tangent space of B to the