Page 116 - Curvature and Homology

98 III. RIEMANNIAN MANIFOLDS: CURVATURE, HOMOLOGY

the circumflex over aj indicating omission of that symbol. We conclude
by linearity that

for any X E T, and the lemma now follows easily.
We have shown that a tangent vector field X on M defines an
endomorphism i(X) of the exterior algebra A(T*) of degree - 1. It is
the unique anti-derivation with the properties:
(i) i(Xy = 0 for every function f on M, and
(ii) i(X)a = (X, a) for every X E T and a E T*.
We remark that i(X) is an anti-derivation whose square vanishes.
This is seen as foliows: i(X)i(X) is a derivation annihilating AP(T*)
for p = 1,2. Hence, since A(T*) is a graded algebra, it is annihilated
by i(X)i(X)-

3.4. Infinitesimal transformations

Relative to the system of local coordinates ul, ..a, un at a point P of the
differentiable manifold M, the contravariant vectors (a/aul),, -..,
form a basis for the tangent space Tp at P. If F denotes the algebra of
differentiable functions on M and f E F, the scalar (af/aui), e(P) is the
directional derivative off at P along the tangent vector Xp at P whose
components in the local coordinates (ui(P)) are given by fl(P), ..-, fn(P).
We define a linear map which is again denoted by X, from F into R :

Evidently, it has the property

In this way, a tangent vector at P may be considered as a linear map of F
into R satisfying equation (3.4.2).
Now, an infinitesimal transformation or vector field X is a map
assigning to each P E M a tangent vector Xp E Tp (cf. 5 1.3). If we define
the function Xf by (Xf) (P) = Xpf for all P E M, the infinitesimal
transformation X may be considered as a linear map of F into the algebra
of all real-valued functions on M with the property