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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2004 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2004 • Question 4

Let \(p\) be a homogeneous polynomial in \(k\) real variables. This means that \[ p(tx_1,tx_2,\cdots ,tx_k)=t^m p(x_1,x_2,\cdots ,x_k) \] for all \(t,x_1,x_2,\cdots ,x_k\in \mathbb {R}\). Prove that for \(a\ne 0\) the set \[ \{x\in \mathbb {R}^k\mid p(x)=a\} \] is a submanifold of \(\mathbb {R}^k\).

Raw LaTeX

Let \(p\) be a homogeneous polynomial in \(k\) real variables. This means that
\[
  p(tx_1,tx_2,\cdots,tx_k)=t^m p(x_1,x_2,\cdots,x_k)
\]
for all \(t,x_1,x_2,\cdots,x_k\in\mathbb{R}\). Prove that for \(a\ne0\) the set
\[
  \{x\in\mathbb{R}^k\mid p(x)=a\}
\]
is a submanifold of \(\mathbb{R}^k\).

Topic Tags

Smooth Manifolds and Embedded Submanifolds