Following Kobayashi and Nomizu, a connection on a manifold is given by a establishing a notion of horizontal vector in the tangent space of a frame bundle. (Alternative approaches make covariant differentiation foundational.)
An important step in developing Riemannian geometry consists of isolating the Levi-Civita connection as that connection with zero torsion that preserves the metric.
Could an alternative approach to defining the Levi-Civita connection go like this: Given a manifold $ M$ with Riemannian metric, construct some natural (family of?) Riemannian metrics on the orthogonal frame bundle of $ M$ . Then simply define “horizontal” to mean orthogonal (in the sense of the constructed metric) to vertical?
Pedagogically, this might offer a bypass around defining and studying torsion.
About my “family of” hedge. There may be no canonical way to compare the scale of vertical vectors, essentially elements of the Lie algebra of the orthogonal group, with more general vectors.
If this is worked out anywhere, I’d appreciate a reference. If there’s some obstruction to this approach, I’d appreciate an explanation.
I do see that this is related to A geometric interpretation of the Levi-Civita connection? and Intuition for Levi-Civita connection? .
I have also asked this on Mathematics Stack Exchange without feeling satisfied by the responses there, as you can see from the comments: https://math.stackexchange.com/questions/2529479/levi-civita-connections-from-metrics-on-the-orthogonal-frame-bundle/2530478?noredirect=1#comment5228620_2530478
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