Page 211 - Curvature and Homology

EXERCISES

2. Establish the identities:
(i) Dx+y = DX + DY,
(ii) D,xY = f (DxY),
(iii) Dx(Y + 2) = DxY + DxZ,
(iv) DfiY) = (Xf)Y +f(DxY),
(v) D~Y = D~Y (if M is almost complex)
for all X, Y, Z E T and f E F-the algebra of differentiable functions on M;

where the rg are the coefficients of the Levi Civita connection.
From (ii) it follows that for any point P, DxY(P) depends only on X(P) and Y,
that is, if Xl(P) = X,(P), then Dxl Y(P) = Dx, Y(P).
3. A p-form a on M may be considered as an alternating multilinear form on
the F-module T with values in F, that is a(Xl, ..., X,) E F for any XI, ..., X, E T.
To a p-form a on M we may associate a p-form Dxa on M called the cowariant
derivative of a with respect to X by putting

Show that the map

so defined is a derivation.
The map Dx may be extended in the obvious way to tensors on M of type
(0,~) which are not necessarily skew-symmetric. Hence, the covariant derivative
of the metric tensor g with respect to the vector field X vanishes, that is
DJ& =o
for all X E T.
4. Establish the equivalence of the following statements for an hermitian
manifold with metric g whose complex structure is defined by J:
(4 WJY) = J(DxY),
(b) DxQ = 0 where Q(X, Y) = g( JX, Y),
(c) dl2 = 0
for any X,Y E T.