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What is the metric good for? It looks like what we need is a connection in E, as well as a connection in the tangent bundle. No metric needed. Also, according to the reference, the second derivative is only symmetric if the curvature of the connection in E vanishes.
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| This article is rated Start-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | ||||||||||
| |||||||||||
What is the metric good for? It looks like what we need is a connection in E, as well as a connection in the tangent bundle. No metric needed. Also, according to the reference, the second derivative is only symmetric if the curvature of the connection in E vanishes.