We prove a Gauss-Bonnet formula for closed semi-algebraic sets.

The Gauss Bonnet theorem links differential geometry with topology. The following expository piece presents a proof of this theorem, building up all of the necessary topological tools. Important applications of this theorem are discussed.

Jolyane Meloche,* Marie Le Guen,* François Potus, Jérôme Vinck, Benoit Ranchoux, Ian Johnson, Fabrice Antigny, Eve Tremblay, Sandra Breuils-Bonnet, Frederic Perros, Steeve Provencher, and Sébastien Bonnet Pulmonary Hypertension and Vascular Biology Research Group from the Quebec Heart and Lung Institute, Department of Medicine, Laval University, Quebec City, Quebec, Canada; and University Paris...

The conditions for the existence and stability of cosmological power-law scaling solutions are established when the Einstein-Hilbert action is modified by the inclusion of a function of the Gauss-Bonnet curvature invariant. The general form of the action that leads to such solutions is determined for the case where the universe is sourced by a barotropic perfect fluid. It is shown by employing ...

We use analytic continuation to derive the Euler-Lagrange equations associated to the Pfaffian in indefinite signature (p, q) directly from the corresponding result in the Riemannian setting. We also use analytic continuation to derive the Chern-Gauss-Bonnet theorem for pseudo-Riemannian manifolds with boundary directly from the corresponding result in the Riemannian setting. Complex metrics on...

Let (X, 0) ⊂ (R, 0) be the germ of a closed subanalytic set and let f and g : (X, 0) → (R, 0) be two subanalytic functions. Under some conditions, we relate the critical points of g on the real Milnor fibre X ∩ f(δ)∩Bǫ, 0 < |δ| ≪ ǫ ≪ 1, to the topology of this fibre and other related subanalytic sets. As an application, when g is a generic linear function, we obtain an “asymptotic” Gauss-Bonnet...

The goal of this paper is to describe Zermelo’s navigation problem on Riemannian manifolds as a time-optimal control problem and give an efficient method in order to evaluate its control curvature. We will show that up to change the Riemannian metric on the manifold the control curvature of Zermelo’s problem has a simple to handle expression which naturally leads to a generalization of the clas...

We give a short proof of the Gauss-Bonnet theorem for a real oriented Riemannian vector bundle E of even rank over a closed compact orientable manifold M . This theorem reduces to the classical Gauss-Bonnet-Chern theorem in the special case when M is a Riemannian manifold and E is the tangent bundle of M endowed with the Levi-Civita connection. The proof is based on an explicit geometric constr...

We obtain an analytic expression for the highly damped asymptotic quasinormal mode frequencies in dimensions d ≥ 5 of the Gauss-Bonnet black hole, which appears in string derived models of gravity. The analytic expression is obtained in the limit when the coupling in front of the Gauss-Bonnet term is small. Although there are several similarities of this geometry with that of the Schwarzschild ...