MAT 531 (Spring 1996) Topology/Geometry II

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.

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Tony Phillips
April 18 EST 1996