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

Abstract

Let M be a smooth manifold and S ⊆ M a properly embedded smooth submanifold. Suppose that we have a fibre metric on TM | s i.e . a positive definite real inner-product on T p M 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.