**Week 12**

1. The Lie derivative along a smooth vectorfield X.

The associated 1-parameter group *phi*_t of diffeomorphisms
can be used to differentiate objects "living on" the
manifold: functions, vectorfields, differential forms and
tensors in general. The symbol for this differentiation
operator is L_X. For example, if f is a smooth real-valued
function,

L_X(f)(x) = lim (1/h)[f(*phi*_h(x)) - f(x)]

(the limit here and elsewhere in this section is as h --> 0).
In this case this is just what we have called Xf before, since
t --> *phi*_t(x) is a curve through x with tangent
vector X(x) at t=0.

The problem with extending this differentiation to a
vectorfield Y is that the analogous expression

[Y(*phi*_h(x)) - Y(x)]

does not make sense, since the two vectors being
subtracted lie in *different* vector spaces. The
remedy is to use the Jacobian map of the diffeomorphisms
to move Y(*phi*_h(x)) into the tangent space at x.
The definition that works is

L_X(Y)(x) = lim (1/h)[(*phi*_{-h})_*Y(*phi*_h(x)) - Y(x)],

since *phi*_{-h} is the inverse of *phi*_h, and
therefore its Jacobian maps the tangent space at *phi*_h(x)
to the tangent space at x.

Similarly, if w is a differential form, we define

L_X(w)(x) = lim (1/h)[(*phi*_{h})^*w(*phi*_h(x)) - w(x)],

since (*phi*_{h})^* pulls back the cotangent space at
*phi*_h(x) to the cotangent space at x.

The behavior of L_X with respect to products of the form
fY, fw, w(Y) is given by the following formulas from
Spivak, Chapter V, Proposition 8:

L_X(fY) = Xf.Y + fL_X(Y)

L_X(fw) = Xf.w + fL_X(w)

L_X(w(Y)) = (L_X(w))(Y) + w(L_X(Y))

which together with the formulas

L_X(dx^i) = sum_1^n Da^i/Dx^j dx^j

L_X(D/Dx^i) = -sum_1^n Da^j/Dx^i D/Dx^j

allow the explicit calculation of L_X(Y) and
L_X(w) in terms of their components in a coordinate system.

2. Singular Homology Theory. (Bredon, Chapter IV).

Standard *p*-simplex, affine singular *p*-simplex,
face maps, singular *p*-simplex, singular *p*-chain,
the singular *p* chain group *Delta*_*p*(*X*)
of a topological space *X*, the boundary homomorphism
*D*_*p*: *Delta*_*p*(*X*) -->
*Delta*_{*p*-1}(*X*). Class exercise: show
that *D*_1 o *D*_2 = 0.

(Back to Week-by-week page)

Tony Phillips

tony@math.sunysb.edu

April 18 EST 1996