MORSE THEORY AND THE GAUSS-BONNET FORMULA Alina ...

T (M, r) = {x ∈ N : there is a geodesic γ of length l(γ) ≤ r from x meeting M
orthogonally}
Weyl [11] showed that, for a submanifold M 2n of R N :
Vol T (M, r) = (πr2 ) (N−2n)/2
((N−2n)/2)!
n�
i=0
k2i(M)r 2i
(N − 2n + 2)(N − 2n + 4)...(N − 2n + 2i)
where the k2i’s are integrals of certain rather complicated curvature functions. For
�
example, k0(M) = Vol M and k2(M) = τdV where τ is the scalar curvature of
M. For our purposes, the most interesting coefficient is the top one, k2n(M). The
connection between this coefficient and the Gauss-Bonnet theorem is provided by the
Pfaffian of a certain matrix of curvature forms, as described below.
M
Let A be a 2n × 2n antisymmetric matrix, with respect to the standard basis
x1, ..., x2n on R 2n . We can define a 2-form ω on R 2n by ω(X, Y ) = 〈AX, Y 〉 for
any vectors X, Y . If dx1, ..., dx2n are the dual 1-forms for the standard basis, then
we define the Pfaffian Pf(A) by ω n = Pf(A)dx1 ∧ ... ∧ dx2n. One can show that
Pf(A) = 1
2 n n!
�
σ∈S2n
ɛσAσ1σ2...Aσ2n−1σ2n where S2n denotes the set of permutations on
2n letters and ɛσ is 1 if σ is an even permutation, −1 otherwise. The Pfaffian also
satisfies Pf(A) 2 = det(A) and Pf(CDC T ) = Pf(D) det(C) for any antisymmetric
matrix D and arbitrary matrix C.
In our case, let {Ei, ..., E2n} be a local orthonormal frame. Define curvature forms
Ωij with respect to this frame as the 2-forms satisfying Ωij(X, Y ) = 〈R(X, Y )Ei, Ej〉
for any vector fields X, Y , where R is the curvature tensor of M. Then the matrix
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