Related papers: Chern's magic form and the Gauss-Bonnet-Chern mass
In this paper, we give a simple proof of the Gauss-Bonnet-Chern theorem for a real oriented Finsler vector bundle with rank equal to the dimension of the base manifold. As an application, a Gauss-Bonnet-Chern formula for any…
We define a generalized mass for asymptotically flat manifolds using some higher order symmetric function of the curvature tensor. This mass is non-negative when the manifold is locally conformally flat and the $\sigma_k$ curvature vanishes…
In this paper, a new proof of the Positive Mass Theorem is established through a newly discovered monotonicity formula, holding along the level sets of the Green's function of an asymptotically flat $3$-manifold. In the same context and for…
We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the…
We revisit the interplay between the mass, the center of mass and the large scale behavior of certain isoperimetric quotients in the setting of asymptotically flat $3$-manifolds (both without and with a non-compact boundary). In the…
In this paper, we develop a general study of contributions at infinity of Bochner-Weitzenb\"ock-type formulas on asymptotically flat manifolds, inspired by Witten's proof of the positive mass theorem. As an application, we show that similar…
The combination of words ``discrete curvature'' is only an apparent contradiction. In this survey we describe curvature notions associated with polygons, polyhedral surfaces, and with abstract polyhedral manifolds. Several theorems about…
We generalise the classical Chern-Gauss-Bonnet formula to a class of 4-dimensional manifolds with finitely many conformally flat ends and singular points. This extends results of Chang-Qing-Yang in the smooth case. Under the assumptions of…
We propose a new definition of the ADM mass for asymptotically Euclidean manifolds inspired by the definition of mass for weakly regular asymptotically hyperbolic manifolds by Gicquaud and Sakovich. This version of the mass allows one to…
We prove Chern conjecture, which states that the Euler characteristic vanishes for closed flat affine manifolds. Our key innovation is a deformation argument for the Euler form.
We prove that a Gaussian ensemble of smooth random sections of a real vector bundle over compact manifold canonically defines a metric on the bundle together with a connection compatible with it. Additionally, we prove a refined…
In this paper, we establish an intrinsic Gauss-Bonnet-Chern formula for Finsler manifolds by using the Mathai-Quillen's superconnection formalism, in which no extra vector field is involved. Furthermore, we prove a more general Lichnerowicz…
It is well known that a physical medium that sets a Lorentz frame generates a Lorentz-breaking gap for a graviton. We examine such generated "mass" terms in the presence of a fluid medium whose ground state spontaneously breaks spatial…
We derive the Chern-Gauss-Bonnet Theorem for manifolds with smooth non-degenerate boundary in the pseudo-Riemannian context from the corresponding result in the Riemannian setting by examining the Euler-Lagrange equations associated to the…
We derive an explicit formula for the ADM mass of asymptotically locally Euclidean (ALE) almost K\"ahler manifolds. The formula expresses the mass in terms of the total Hermitian scalar curvature and topological data associated with the…
Inspired by a formula of Stern that relates scalar curvature to harmonic functions, we evaluate the mass of an asymptotically flat $3$-manifold along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral…
An explicit lower bound for the mass of an asymptotically flat Riemannian 3-manifold is given in terms of linear growth harmonic functions and scalar curvature. As a consequence, a new proof of the positive mass theorem is achieved in…
Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal O_{X_k}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth…
% A new, formal, non-combinatorial approach to invariants of % three-dimensional manifolds of Reshetikhin, Turaev and % Witten in the framework of non-perturbative topological % quantum Chern-Simons theory, corresponding to an arbitrary %…
On any odd-dimensional oriented Riemannian manifold we define a volume form, which we call the odd Pfaffian, through a certain invariant polynomial with integral coefficients in the curvature tensor. We prove an intrinsic Chern-Gauss-Bonnet…