Page 36 - Curvature and Homology

18 I. RIEMANNIAN MANIFOLDS
where the are the (contravariant) components of X in the local
coordinates ul, ..., un. This, in turn is equal to the directional derivative
off' along the contravariant vector

at 4P). By mapping X in Tp into X' in Tp, we get a linear map of
Tp into THPt This is the induced map 4,. The map 4 is said to be regulm
(at P) if the induced map +, is 1 - 1.
A subset M' of M is called a submanifold of M if it is itself a differenti-
able manifold, and if the injection +' of M' into M is a regular
differentiable map. When necessary we shall denote M' by (+', M').
Obviously, we have dim M' I; dim M. The topology of M' need not
coincide with that induced by M on M'. If M' is an open subset of M,
then it possesses a naturally induced differentiable structure. In this case,
M' is called an open subma&fold of M.
Recalling the definition of regular surface we see that the above
univalence condition is equivalent to the condition that the Jacobian
of 4 is of rank n.
By a clbsed submanifold of dimension r is meant a submanifold M' with
the properties: (i) 4'(Mf) is closed in M and (ii) every point P E +'(Mf)
belongs to a coordinate neighborhood U with the local coordinates
ul, ..., un such that the set +'(Mf) n U is defined by the equations
ur+l = 0, ..., un = 0. The definition of a regular closed surface given
in 5 1.1 may be included in the definition of closed submanifold.
We shall require the following notion: A parametrized curwe in M
is a differentiable map of class k of a connected open interval of R into M.
The differentiable map + : M -+ M' induces a map +* called the dual
of 4, defined as follows:

The map +* may be extended to a map which we again denote by +*

as follows: Consider the pairing (vl A ... A vi, w: A ... A w:) defined by

A
(~1 ... A v,, w,* A ... A w,*) = p! det ((v,, w:)) (1.5.1)