Page 35 - Curvature and Homology

1.5. SUBMANIFOLDS

and (ii)' is equivalent to
(ii)"
where agl...,p is a p-vector.
The condition (ii)" shows that the Kronecker symbol is actually a
tensor of type (p, p).
Now, let
a = a(il...ip) duil A ... A duip

Then,
a A P = c~,...~,,, duil A ... A duh+q
where

and
j(i l...ip) "(il...~
(P + I)! da = Skl...kp+l -
au j dukl A ... A dukp+l. (1.4.1 1)
From (1.4.10) we deduce

1.5. Submanifolds

The set of differentiable functions F (of class k) in a differentiable
manifold M (of class k) forms an algebra over R with the usual rules
of addition, multiplication and scalar multiplication by elements of R.
Given two differentiable manifolds M and M', a map 4 of M into M'
is called differentiable, if f' . 4 is a differentiable function in M for
every such function f' in M'. This may be expressed in terms of local
coordinates in the following manner: Let ul, ..., un be local coordinates
at P E M and vl, ..., vm local coordinates at +(P) E M'. Then 4 is a
differentiable map, if and only if, the vi(+(ul, ..., un)) = vi(ul, ..., un) are
differentiable functions of ul, ..., un. The map 4 induces a (linear)
differentiable map 4, of the tangent space Tp at P E M into the tangent
space TH4 at P' = #(P) E M'. Let X E Tp and consider a differentiable
function f' in the algebra F' of differentiable functions in M'. The
directional derivative off' 4 along X is given by