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Higher transgressions of the Pfaffian. (English) Zbl 1504.58003

The classical Gauss-Bonnet theorem reads \[ 2\pi\chi(M)=\int_{M}\mathfrak{k}_{g}\upsilon_{g}-\int_{\partial M}a\cdot l_{g}+\sum_{p}\angle^{\mathrm{out}}(p) \] where \((M,g)\) is a closed surface with boundary \(\partial M\) and corners \(p\), \(\mathfrak{k}_{g}\) is the Gaussian curvature, \(a:\partial M\hookrightarrow M\) is the geodesic curvature function with respect to the outer normal, \(\upsilon_{g}\) is the volume density, \(l_{g} \) is the length element on \(\partial M\), and \(\angle^{\mathrm{out}}(p)\) is the outer angle at a corner \(p\).
The principal objective in this paper is to extend the Gauss-Bonnet formula of C. B. Allendoerfer and A. Weil [Trans. Am. Math. Soc. 53, 101–129 (1943; Zbl 0060.38102)] to general compact Riemannian polyhedral manifolds (Theorem 6.1 and Theorem 6.5). A strategy of proof of the Gauss-Bonnet formula on polyheral manifolds similar to this paper was proposed in [P. Wintgen, Colloq. Math. Soc. Janos Bolyai 31, 805–816 (1982; Zbl 0509.53037)].
The synopsis of the paper goes as follows.
§2 and §3 review the Pffafian of the curvature using the language of double terms.
§4 defines smooth polyhedral manifolds and polyhedral complexes, studying their properties with respect to integration of forms. The category of polyhedral complexes allows of bundling together the outer cones of faces of Riemannian polyhedral manifolds, the natural locally trivial bundles of spherical types where the contributions of the faces are localized.
§5 starts with Chern’s construction of a transgression form [S.-S. Chern, Ann. Math. (2) 45, 747–752 (1944; Zbl 0060.38103)], introducing higher transgressions for the Pfaffian form on vector bundles rigged out in a nondegerate bilinear form and a compatible connection. It is shown that the exterior differential of these transgressions can be computed as a sum of lower-order transgressions.
§6 applies the abstract transgression theorem to the case of Riemannian polyhedral manifolds. The formula has been obtained with entirely different methods by Allendoerfer and Weil [loc. cit.] for a particular class of regular polyhedral manifolds. For regular polyhedral manifolds, the Gauss-Bonnet formula in the even-dimensional case follows by iterating the transgression formula on the boundary strata. The general case can be dealt with by using a global polyhedral complex to transfer
§7 particularizes the formula to space forms.
Theorem. Let \(M\) be a \(d\)-dimensional compact polyhedral manifold of constant sectional curvature \(\mathfrak{k}\), with totally geodesic faces. Then \[ \frac{\varkappa(M)}{2}=\sum_{j\geq0}\sum_{Y\in\mathcal{F} ^{(d-2j)}(M)}\mathfrak{k}^{j}\frac {\mathrm{vol}_{2j}(Y)}{\mathrm{vol}(S^{2j})}\frac{\angle^{\mathrm{out}}Y}{\mathrm{vol}(S^{d-2j-1})} \] where \(\mathcal{F}^{(d-2j)}(M)\) is the set of faces of of dimension \(2j\), \(S^{k}\) is the standard unit sphere in \(\mathbb{R}^{k+1}\), and \(\angle^{\mathrm{out}}Y\) is the measure of the outer solid angle at the face \(Y\).
The author deduces from this theorem identities for hyperbolic polyhedra involving the volumes of even-dimensional faces and their outer angles, including an extension to the noncompact case where some or all vertices are ideal.

MSC:

58A10 Differential forms in global analysis
58A35 Stratified sets
58A17 Pfaffian systems
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References:

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