Characterizing the Tangent and Normal Bundles - Submanifolds in Banach Spaces and Their Classifications

  1. Let $M\subset\mathbb{R}^n$ of class $C^k$ and dimension $m$. Show that the set $TM=\{(p,v)\in\mathbb{R}^n\times\mathbb{R}^n;p\in M, v\in T_pM\}$ is a submanifold of class $C^{k-1}$ and dimension $2m$ , it is called the Tangent bundle of $M$.
  2. With the notation above, let $vM=\{(p,v)\in\mathbb{R}^n\times\mathbb{R}^n;p\in M, v\in T_pM^\perp\}$. Show that $vM$ is a submanifold of class $C^{k-1}$ and dimension $n$, it is called the Normal bundle of $M$ in $\mathbb{R}^n$.


This is a question in advanced calculus class in the context of Submanifolds in Banach spaces.
Calculus Manifolds Banach Space
Elviegem Elviegem
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