Characterizing the Tangent and Normal Bundles - Submanifolds in Banach Spaces and Their Classifications
- 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 $M$.
- 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.
Elviegem
41
The answer is accepted.
Join Matchmaticians Affiliate Marketing
Program to earn up to a 50% commission on every question that your affiliated users ask or answer.
- accepted
- 433 views
- $60.00
Related Questions
- Find value of cos(2x)
- Parametric, Polar, and Vector-Valued Equations for Kav10
- Explain parameter elimination for complex curves
- Calculus - functions, method of Least Squares
- How do you prove integration gives the area under a curve?
- Fourier series
- Show that the MLE for $\sum_{i=1}^{n}\left(\ln{2x_i} - 2\ln{\lambda} - \left(\frac{x_i}{\lambda}\right)^2\right)$ is $\hat{\lambda} = \sqrt{\sum_{i=1}^{n}\frac{x_i^2}{n}}$.
- Evaluate$\int \sqrt{\tan x}dx$