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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2005 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2005 • Question 2

Let \(V,W:\mathbb {R}^4\to \mathbb {R}^4\) be vector fields on \(\mathbb {R}^4\) defined by \[ V(x,y,z,w)=(y,-x,w,-z) \] and \[ W(x,y,z,w)=(w,z,-y,-x). \] Is there a nonempty surface \(\Sigma \subset \mathbb {R}^4\) such that for every \(p\in \Sigma \), we have \(V(p),W(p)\in T_p\Sigma \)? Find such a surface or prove that it does not exist.

Raw LaTeX

Let \(V,W:\mathbb{R}^4\to\mathbb{R}^4\) be vector fields on \(\mathbb{R}^4\) defined
by
\[
  V(x,y,z,w)=(y,-x,w,-z)
\]
and
\[
  W(x,y,z,w)=(w,z,-y,-x).
\]
Is there a nonempty surface \(\Sigma\subset\mathbb{R}^4\) such that for every
\(p\in\Sigma\), we have \(V(p),W(p)\in T_p\Sigma\)? Find such a surface or prove that it
does not exist.

Topic Tags

Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups Vector Fields and Flows