A new proof of a classical result on the topology of orientable connected and compact surfaces by means of the Bochner technique

Autor:	Almira, J. M., Romero, A.
Rok vydání:	2019
Předmět:	Mathematics - Differential Geometry
Druh dokumentu:	Working Paper
Popis:	As an application of the Bochner formula, we prove that if a $2$-dimensional Riemannian manifold admits a non-trivial smooth tangent vector field $X$ then its Gauss curvature is the divergence of a tangent vector field, constructed from $X$, defined on the open subset out the zeroes of $X$. Thanks to the Whitney embedding theorem and a standard approximation procedure, as a consequence, we give a new proof of the following well-known fact: if on an orientable, connected and compact $2$-dimensional smooth manifold there exists a continuous tangent vector field with no zeroes, then the manifold is diffeomorphic (or equivalently homeomorphic) to a torus. Comment: 6 pages, submitted to a journal
Databáze:	arXiv
Externí odkaz:	http://arxiv.org/abs/1911.08821 Zobrazit plný text záznamu View this record from Arxiv