Mathematics and Statistics Vol. 7(4), pp. 146 - 149
DOI: 10.13189/ms.2019.070408
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A Note on Locally Metric Connections

Mihail Cocos ^*
Department of Mathematics, Weber State University, Ogden, UT 84408, USA

ABSTRACT

The Fundamental Theorem of Riemannian geometry states that on a Riemannian manifold there exist a unique symmetric connection compatible with the metric tensor. There are numerous examples of connections that even locally do not admit any compatible metrics. A very important class of symmetric connections in the tangent bundle of a certain manifolds (afinnely flat) are the ones for which the curvature tensor vanishes. Those connections are locally metric. S.S. Chern conjectured that the Euler characteristic of an affinely at manifold is zero. A possible proof of this long outstanding conjecture of S.S. Chern would be by verifying that the space of locally metric connections is path connected. In order to do so one needs to have practical criteria for the metrizability of a connection. In this paper, we give necessary and sufficient conditions for a connection in a plane bundle above a surface to be locally metric. These conditions are easy to be veri ed using any local frame. Also, as a global result we give a necessary condition for two connections to be metric equivalent in terms of their Euler class.

KEYWORDS
Affine Connections, Locally Metric Connections, Affinely Flat Manifolds, Euler Class of a Connection

Cite This Paper in IEEE or APA Citation Styles
(a). IEEE Format:
[1] Mihail Cocos , "A Note on Locally Metric Connections," Mathematics and Statistics, Vol. 7, No. 4, pp. 146 - 149, 2019. DOI: 10.13189/ms.2019.070408.

(b). APA Format:
Mihail Cocos (2019). A Note on Locally Metric Connections. Mathematics and Statistics, 7(4), 146 - 149. DOI: 10.13189/ms.2019.070408.