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Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2020 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2020 • Question 1

Show that \(\mathcal {C}\cap S\) is an embedded submanifold in \(\mathbb {R}^m\times \mathbb {R}^n\) and find its dimension, where \[ \mathcal {C}=\{(x,y)\in \mathbb {R}^m\times \mathbb {R}^n:|x|^2=|y|^2\},\qquad S=\{(x,y)\in \mathbb {R}^m\times \mathbb {R}^n:|x|^2+|y|^2=1\}. \]

Raw LaTeX

Show that \(\mathcal{C}\cap S\) is an embedded submanifold in
\(\R^m\times\R^n\) and find its dimension, where
\[
  \mathcal{C}=\{(x,y)\in\R^m\times\R^n:|x|^2=|y|^2\},\qquad
  S=\{(x,y)\in\R^m\times\R^n:|x|^2+|y|^2=1\}.
\]

Topic Tags

Smooth Manifolds and Embedded Submanifolds