metric independent).Ĭonsider a smooth manifold with boundary $M$. The XSection application is ideal for creating, editing, and printing of geological cross sections. Let $P(M)$ be the Pfaffian integral over $M$ (which weĪre assuming is a topological invariant, i.e. Once you've checked this, then it follows that it is Same axioms as the Euler characteristic (up to an overall multiplicationīy a constant). One needs to check that the topological invariant it defines is theĮuler characteristic. (this is certainly true in the proof of the classic Gauss-Bonnet theorem).Īssuming topological invariance (independence of the metric), then This is, but I think it can be boiled down to an application of Stokes theorem The integral on the left does not change. So one needs to see thatįor a 1-parameter family of metrics on $M$ (or corresponding connections), It should be independent of the Riemannian metric. The first is to verify that it is a topological invariant. One may break down the generalized Gauss-Bonnet into two parts. Specializing further, if $M$ is a compact complex manifold of dimension $k$ and $E=T_M$ its holomorphic tangent bundle, then Where $\Theta(E,\nabla)$ is the curvature of the connection $\nabla$.
If $\sigma$ is a smooth section of $E$ having non-degenerate zeros, then the integer $c_k(E)$ counts precisely these zeros (with sign, depending on orientations), which compose the degeneracy cycle $D_1$.įinally, to recover the top Chern class of $E$ from its differential-geometric data, just recall that the Chern forms $c_r(E,\nabla)$ of a vector bundle $E$ endowed with a connection $\nabla$ are defined by the formula When $E\to M$ is an oriented vector bundle of rank $2n$ over aĬompact manifold $M$, it has a well-defined de Rham Euler class $e(E)$
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