By James R. Milgram
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Extra resources for Algebraic and Geometric Topology
L on forms w, Wi E fY(M, Q). For any connection V' in Q and its associated exterior differentiation d\7 in n"(M, Q), there is a codifferential 15\7 : nr+1(M, Q) ~ fY(M, Q). It is given by 15\7W = (_1)nr+1 * d\7 * w. The evaluation formula for an orthonormal frame E 1 , ... ,En is as follows: n 15\7W = - L iEA V'EA W. A=l For this operator to be formally the adjoint of d\7, more compatibility conditions for V' and 9 are required. We leave this question aside for the moment, and evaluate the operator 15\7 on 7r E n1 (M, Q).
61) is well-defined as a vector field along ,. The same interpretation has to be given to several expressions in the calculation to follow. Since V is F-Jacobi along the geodesic " we have 7r~(V'rV'rV + RM (V, i'h) = 7r~(V'rV't1i' + V'rb, V]- V'~,"rli' - V'rV'ifi') = o.
G. 17) a flow of isometries. Associated to a : L 181 L ----t Q there is a shape operator or Weingarten map. 18) gQ(a(U, V),s) = g(W(s)U, V) for U, V E fL. Let 7r~ : TM ----t L denote the orthogonal projection corresponding to the decomposition TM = L ffi L~ (L~ ~ Q). 19) Thus the characteristic polynomial of W is a geometric invariant associated to:F. Of particular interest is TrW(s). It is linear in s, hence TrW E fQ*. 20) K(V) = 0 for V E f L, K(S) = Tr W(s) for s E fQ, where we have used the identification L~ ~ Q.
Algebraic and Geometric Topology by James R. Milgram
Categories: Algebraic Geometry