(a) Let M be a Cr submanifold of Rn, and let f : M → R...

(a) Let M be a Cr submanifold of Rn, and let f : M → R be a Cr function. Show

there is an open neighbourhood V of M in Rn and a Cr function g : V → R such

that f = g|M.

(b) Show that, if M is a closed subset of Rn, then we can take V = Rn

.

(c) Can we take V = Rn in general? Why or why not?

We've just learned partition of unity ,please don't use any high-level therom!!

Expert Solution

Answer:-

Consider a subset M of RN is a k-dimensional submanifold if for every point x in M there exists a neighbourhood V of x in RN , an open set U ⊆ Rk, and a smooth map ξ : U → RN such that ξ is a homeomorphism onto M ∩ V , and Dyξ is injective for every y ∈ U.

Colby Messinger answered 2 years ago

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Question

(a) Let M be a Cr submanifold of Rn, and let f : M → R...

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