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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2009 • Question 1

Question

Math 6510 • Differentiable Manifolds Spring 2009 • Question 1

Let \(f:\mathbb {R}^3\times \mathbb {R}^3\to \mathbb {R}^2\) be defined by \(f(x,y)=(x\cdot y, |x|^2-|y|^2)\), where \(x\cdot y\) is the dot product of \(x\) and \(y\) in \(\mathbb {R}^3\) and \(|x|\) is the Euclidean norm of \(x\). Suppose that \((a,b)\ne (0,0)\). Show that \[ \{(x,y)\in \mathbb {R}^3\times \mathbb {R}^3:f(x,y)=(a,b)\} \] is a smooth submanifold, and compute it’s dimension.

Raw LaTeX

Let \(f:\R^3\times\R^3\to\R^2\) be defined by
\(f(x,y)=(x\cdot y, |x|^2-|y|^2)\), where \(x\cdot y\) is the dot product of \(x\) and
\(y\) in \(\R^3\) and \(|x|\) is the Euclidean norm of \(x\). Suppose that
\((a,b)\ne(0,0)\). Show that
\[
  \{(x,y)\in\R^3\times\R^3:f(x,y)=(a,b)\}
\]
is a smooth submanifold, and compute it's dimension.

Topic Tags

Cohomology and Cup Products Pullbacks and Wedge Products Smooth Manifolds and Embedded Submanifolds