I am trying to learn differential geometry on my own, and I'm finally getting serious about learning the subject. I am using several books which I supplement with material I find on the internet. I have found some excellent lecture notes on differential geometry written by Dmitri Zaisev. I like his discussion of tangent vectors from several viewpoints. My question concerns a definition that I have found in those notes.(adsbygoogle = window.adsbygoogle || []).push({});

The definition of asubmanifoldis given as:

A subset S ⊂ ℝ^{n}is an m-dimensional submanifold of ℝ^{n}(m < n) of class C^{k}if ∀ p in S ∃ an open neighborhood V_{p}⊂ ℝ^{n}, an open set W_{p}⊂ ℝ^{m}, and a homeomorphism f_{p}: W_{p}→ V_{p}∩ S which is C^{k}and regular in the sense that ∀ a ∈ W_{p}, the differential d_{a}f_{p}: ℝ^{m}→ ℝ^{n}is injective.

I need help pulling this definition apart. In particular, I am perplexed as to the part the set W_{p}plays. If we need an open set in S, why isn't V_{p}∩ S an open set in S? Why do we need f_{p}to map from W_{p}to V_{p}∩ S because it seems to me that they are both sets contained in S.

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# Submanifold definition

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