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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2019 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 2019 • Question 1

Let \(M^m\subset \mathbb {R}^n\) be a smooth embedded submanifold. Show that the unit tangent bundle \[ \UM =\{(p,V):p\in M\text { and }V\in T_pM\text { such that }|V|=1\} \] is a smooth embedded submanifold of \(T\mathbb {R}^n\approx \mathbb {R}^n\times \mathbb {R}^n\). What is its dimension?

Raw LaTeX

Let \(M^m\subset\R^n\) be a smooth embedded submanifold. Show that the unit
tangent bundle
\[
  \UM=\{(p,V):p\in M\text{ and }V\in T_pM\text{ such that }|V|=1\}
\]
is a smooth embedded submanifold of \(T\R^n\approx\R^n\times\R^n\). What is its
dimension?

Topic Tags

Smooth Manifolds and Embedded Submanifolds Tangent Bundles and Vector Bundles