A new proof of Huber's theorem on differential geometry in the large

微分流形上微分形式的Gaffney不等式新证明：Kozono-Yanagisawa方法的变分方法

A new proof of the Gaffney's inequality for differential forms on manifolds-with-boundary: the variational approach à la Kozono--Yanagisawa

本文通过变分方法和Bochner技术，为微分流形上微分形式的Gaffney不等式提供新证明。

Intrinsic Differential Geometry and the Existence of Quasimeromorphic Mappings

We give a new proof of the existence of nontrivial quasimeromorphic mappings on a smooth Riemannian manifold, using solely the intrinsic geometry of the manifold.

在Lean中合成微分几何

Synthetic Differential Geometry in Lean

本文用Lean证明助手和mathlib库形式化合成微分几何，证明了多变量函数的泰勒定理，展示了mathlib在构造数学中的潜力。

Positive Complex Sectional Curvature, Ricci Flow and the Differential Sphere Theorem

The paper provides a different proof of the result of Brendle-Schoen on the differential sphere theorem. It is shown directly that the invariant cone of curvature operators with positive (or non-negative) complex sectional curvature is preserved by the Ricci flow. This implies, by a result of Böhm-Wilking, that the normalized Ricci flow deforms such a metric to a metric of constant positive curvature. Using earlier work of Yau and Zheng it can be shown that a metric with strictly (pointwise) 1/4-pinched sectional curvature has positive complex sectional curvature. This gives a direct proof of …

A Proof of the Isometric Embedding Theorem in Three Dimensional Euclidean Space

A proof of the isometric embedding of a given two-metric in E^3 of class C^1. The method uses the theory of first order partial differential equations. The curvature of the metric plays no role in the proof.

Elementary Proofs Of The Gauss-Bonnet Theorem And Other Integral Formulas In $\Bbb R^3$

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.

Lagrangian mechanics without ordinary differential equations

A variational proof is provided of the existence and uniqueness of evolutions of regular Lagrangian systems.

Novikov type inequalities for differential forms with non-isolated zeros

We generalize the Novikov inequalities for 1-forms in two different directions: first, we allow non-isolated critical points (assuming that they are non-degenerate in the sense of R.Bott), and, secondly, we strengthen the inequalities by means of twisting by an arbitrary flat bundle. The proof uses Bismut's modification of the Witten deformation of the de Rham complex; it is based on an explicit estimate on the lower part of the spectrum of the corresponding Laplacian. In particular, we obtain a new analytic proof of the degenerate Morse inequalities of Bott.

Differential Rigidity of Anosov Actions of Higher Rank Abelian Groups and Algebraic Lattice Actions

We show that most homogeneous Anosov actions of higher rank Abelian groups are locally smoothly rigid (up to an automorphism). This result is the main part in the proof of local smooth rigidity for two very different types of algebraic actions of irreducible lattices in higher rank semisimple Lie groups: (i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the actions of cocompact lattices on Furstenberg boundaries, in particular, projective spaces. The main new technical ingredient in the proofs is the use of a proper "non-stationary" generalization of the classical the…