Dirk Siersma, Mihai Tibar
The total curvature of complex hypersurfaces in $\bC^{n+1}$ and its variation in families appear to depend not only on singularities but also on the behaviour in the neighbourhood of infinity. We find the asymptotic loss of total curvature towards infinity and we express the total curvature and the Gauss-Bonnet defect in terms of singularities and tangencies at infinity.
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In this work, we prove a version of the fundamental theorem of submanifolds to target manifolds with warped structure.
The problem of determining the {\it Bonnet hypersurfaces in} $R^{n+1}$, for $n>1$, is studied here. These hypersurfaces are by definition those that can be isometrically mapped to another hypersurface or to itself (as locus) by at least one nontrivial isometry preserving the mean curvature. The other hypersurface and/or (the locus of) itself is called {\it Bonnet associate} of the initial hypersurface. The orthogonal net which is called \hbox{\it $A$-net} is special and very important for our study and it is described on a hypersurface. It is proved that, non-minimal hypersurface in $R^{n+1}$ …
We deal with hypersurfaces in the framework of the relative differential geometry in $\mathbb{R}^4$. We consider a hypersurface $\varPhi$ in $\mathbb{R}^4$ with position vector field $\vect{x}$ which is relatively normalized by a relative normalization $\vect{y}$. Then $\vect{y}$ is also a relative normalization of every member of the one-parameter family $\mathcal{F}$ of hypersurfaces $\varPhi_μ$ with position vector field $\vect{x}_μ= \vect{x} + μ\, \vect{y}$, where $μ$ is a real constant. We call every hypersurface $\varPhi_μ\in \mathcal{F}$ relatively parallel to $\varPhi$. This considerat…
We study the Bonnet problem for surfaces in 4-dimensional space forms, where two isometric surfaces have the same mean curvature if there exists a parallel vector bundle isometry between their normal bundles that preserves the mean curvature vector fields. We deal with the moduli space of congruence classes of isometric surfaces with the same mean curvature, and with properties inherited on a surface by its structure. The study of this problem led us to a new conformally invariant property, called isotropic isothermicity, that coincides with the usual concept of isothermicity for surfaces lyin…
The aim of this note is to provide an intrinsic proof of the Gauss--Bonnet theorem without invoking triangulations, which is achieved by exploiting complex structures.
Extrinsic Bonnet-Myers Theorem and almost rigidity
本文证明了正Ricci曲率紧致流形的外在Bonnet-Myers定理，并给出了欧氏空间中正截面曲率且外在直径几乎最大的紧致超曲面的几乎刚性结果。
From the point of view of index theory, we give a simple proof of a Gauss-Bonnet-Chern formula for all Finsler manifolds by the Cartan connection. Based on this, we establish a Gauss-Bonnet-Chern formula for any metric-compatible connection and also derive the Gauss-Bonnet-Chern formula of Lackey.
In this paper, we establish a Gauss-Bonnet-Chern theorem for general closed complex Finsler manifolds.
We prove a Gauss-Bonnet theorem for (finite coverings of) moduli spaces of Riemann surfaces endowed with the McMullen metric. The proof uses properties of an exhaustion of moduli spaces by compact submanifolds with corners and the Gauss-Bonnet formula of Allendoerfer and Weil for Riemannian polyhedra.
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 metric-compatible connection is established on Finsler manifolds.
For a compact differentiable surface with boundary embedded in $\Bbb R^3$, we give simple proofs of the Gauss-Bonnet theorem, Poincaré-Hopf theorem, and several other integral formulas. We complete all of the proofs without using fundamental or differential forms.
Gauss-Bonnet formula for metrics with logarithmic singularities on compact Riemann surfaces
该研究在高斯曲率关于度量面积形式Lebesgue可积的条件下，推广了紧黎曼曲面上带对数奇异性度量的Gauss-Bonnet公式，并将其应用于关联三次微分亚纯的特殊Kähler度量。
We give a stereological version of the Gauss-Bonnet formula in order to compute the Euler characteristic of a domain with boundary in a smooth orientable surface in R^3, by looking at contacts with a "sweeping" plane.
Given a construction of smooth homotopy class invariants of smooth immersions $M^n\to R^{n+k}$. The particular case of $k=1, n\ge 1$ is a sequence of non-zero integrals, where the $n=2$ term is the Gauss-Bonnet integral
We prove a Gauss-Bonnet type formula for Riemann-Finsler surfaces of non-constant indicatrix volume and with regular piecewise smooth boundary. We give a Hadamard type theorem for N-parallels of a Landsberg surface.
In this paper, we prove a positive mass theorem and Penrose-type inequality of the Gauss-Bonnet-Chern mass $m_2$ for the graphic manifold with flat normal bundle.
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 construction of the Thom class for 2-plane bundles.
We prove a version of Gauss-Bonnet theorem in sub-Riemannian Heisenberg space $H^1$. The sub-Riemannian distance makes $H^1$ a metric space and consenquently with a spherical Hausdorff measure. Using this measure, we define a Gaussian curvature at points of a surface S where the sub-Riemannian distribution is transverse to the tangent space of S. If all points of S have this property, we prove a Gauss-Bonnet formula and for compact surfaces (which are topologically a torus) we obtain $\int_S K = 0$.
We prove a discrete Gauss-Bonnet-Chern theorem which states where summing the curvature over all vertices of a finite graph G=(V,E) gives the Euler characteristic of G.
As an interesting application of the Einstein-Gauss-Bonnet theory and our work on the Gauss-Bonnet-Chern mass (Ge, Wang, Wu), we obtain a positive mass theorem for asymptotically flat graphs in $\R^{n+1}$ under a condition that $R+αL_2$ is non-negative, where $R$ is the scalar curvature, $α\in\R$ a constant and $L_2$ the second Gauss-Bonnet curvature. A Penrose type inequality is also obtained in the case $α>0$.
In this paper we show positive mass theorems and Penrose type inequalities for the Gauss-Bonnet-Chern mass, which was introduced recently in \cite{GWW}, for asymptotically flat CF manifolds and its rigidity.
It is shown in the paper "Variational Properties of the Gauss-Bonnet Curvatures" of M.L. Labbi, that metrics with constant 2k-Gauss-Bonnet curvature on a closed n-dimensional manifold, 1<2k<n, are critical points for a certain Hilbert type functional with respect to volume preserving conformal variations. This motivates the corresponding Yamabe problem: is it true that any metric on a closed manifold is conformal to a metric with constant 2k-Gauss-Bonnet curvature? Using perturbative methods we affirmatively answer this question for small perturbations of certain space forms. More precisely, i…
In this note, we use Chern's magic form $Φ_k$ in his famous proof of the Gauss-Bonnet theorem to define a mass for asymptotically flat manifolds. It turns out that the new defined mass is equivalent to the one that we introduced recently by using the Gauss-Bonnet-Chern curvature $L_k$. Moreover, this equivalence implies a simple proof of the equivalence between the ADM mass and the intrinsically defined mass via the Ricci tensor, which was reconsidered by Miao-Tam \cite{MT} and Herzlich \cite{H} very recently.
The authors Balogh-Tyson-Vecchi in arXiv:1604.00180 utilize the Riemannian approximations scheme $(\mathbb H^1,<,>_L)$, in the Heisenberg group, introduced by Gromov, to calculate the limits of Gaussian and normal curvatures defined on surfaces of $\mathbb H^1$ when $L\rightarrow\infty$. They show that these limits exist (unlike the limit of Riemannian surface area form or length form), and they obtain Gauss-Bonnet theorem in $\mathbb H^1$ as limit of Gauss-Bonnet theorems in $(\mathbb H^1,<,>_L)$ when $L$ goes to infinity. This construction was extended by Wang-Wei in arXiv:1912.00302 to the …
We give an explicit formula for the Gauss-Bonnet-Chern mass of an asymptotically flat graphical manifold of arbitrary codimension and use it to prove the positive mass theorem and the Penrose inequality for graphs with flat normal bundle.
We define a new one form H^A based on the second fundamental tensor H^abA, the Gauss-Bonnet-Chern form can be novelly expressed with this one-form. Using the phi-mapping theory we find that the Gauss-Bonnet-Chern density can be expressed in terms of the delta-function and the relationship between the Gauss-Bonnet-Chern theorem and Hopf-Poincare theorem is given straightforwardly. The topological current of the Gauss-Bonnet-Chern theorem and its topological structure are discussed in details. At last, the Morse theory formula of the Euler characteristic is generalized.
In this paper we first prove that every differential character can be represented by differential form with singularities. Then we lift the Gauss-Bonnet-Chern theorem for vector bundles to differential characters.
In this paper, we prove a local index theorem for the DeRham Hodge-laplacian which is defined by the connection compatible with metric. This connection need not be the Levi-Civita connection. When the connection is Levi-Civita connection, this is the classical local Gauss-Bonnet-Chern theorem.
We prove the Chern-Gauss-Bonnet Theorem using sigma models whose source supermanifolds have super dimension 0|2. Along the way we develop machinery for understanding manifold invariants encoded by families of 0|n-dimensional Euclidean field theories and their quantization.
By proving graph theoretical versions of Green-Stokes, Gauss-Bonnet and Poincare-Hopf, core ideas of undergraduate mathematics can be illustrated in a simple graph theoretical setting. In this pedagogical exposition we present the main proofs on a single page and add illustrations. While discrete Stokes is is old, the other two results for graphs were found only recently.
The $(2k)$-th Gauss-Bonnet curvature is a generalization to higher dimensions of the $(2k)$-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for $k=1$. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metric…
The Gauss-Bonnet curvature of order $2k$ is a generalization to higher dimensions of the Gauss-Bonnet integrand in dimension $2k$, as the usual scalar curvature generalizes the two dimensional Gauss-Bonnet integrand. In this paper, we evaluate the first variation of the integrals of these curvatures seen as functionals on the space of all Riemannian metrics on the manifold under consideration. An important property of this derivative is that it depends only on the curvature tensor and not on its covariant derivatives. We show that the critical points of this functional once restricted to metri…
We present a self-contained proof of the Gauss-Bonnet theorem for two-dimensional surfaces embedded in $R^3$ using just classical vector calculus. The exposition should be accessible to advanced undergraduate and non-expert graduate students. It may be viewed as an illustration and exercise in multivariate calculus and a motivation to go deeper into the fields of geometry and topology.
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