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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2000 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2000 • Question 4

A. Let \[ S=\{(z_1,\ldots ,z_n)\in \mathbb {C}^n:z_1^2+\cdots +z_n^2=1\}. \] Prove that \(S\) is a smooth (real) submanifold of \(\mathbb {C}^n=\mathbb {R}^{2n}\), and find it’s dimension.

B. Prove that the orthogonal group \(O(n)\) is a smooth submanifold of the space of \(n\times n\) matrices \(M(n)=\mathbb {R}^{n^2}\), and find it’s dimension.

Raw LaTeX

\textbf{A.} Let
\[
  S=\{(z_1,\ldots,z_n)\in\mathbb{C}^n:z_1^2+\cdots+z_n^2=1\}.
\]
Prove that \(S\) is a smooth (real) submanifold of \(\mathbb{C}^n=\mathbb{R}^{2n}\), and
find it's dimension.

\textbf{B.} Prove that the orthogonal group \(O(n)\) is a smooth submanifold of the
space of \(n\times n\) matrices \(M(n)=\mathbb{R}^{n^2}\), and find it's dimension.

Topic Tags

Lie Groups and Matrix Lie Groups Smooth Manifolds and Embedded Submanifolds