A Discussion on the Establishment That a Fibre Metric on the Positive Definite Real Inner-Product of a Properly Embedded Smooth Submanifold Be Always Extended to a Riemannian Metric on the Positive Definite Real Inner-Product
Let M be a smooth manifold and S ⊆M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM|si.e. a positive definite real inner-product on TpM for all p ∈ S, which depends smoothly on p ∈ S. The purpose of this article is to figure out that the fibre metric on TM|s can always be extended to a Riemannian metric on TM from a special perspective.
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