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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2024 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2024 • Question 4

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 \(S\subset \mathbb {R}^4\) such that for every \(p\in S\) both \(V(p)\) and \(W(p)\) are tangent to \(S\)? Find such a surface or show 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 \(S\subset\mathbb{R}^4\) such that for every \(p\in S\) both \(V(p)\) and \(W(p)\) are
tangent to \(S\)? Find such a surface or show it does not exist.

Topic Tags

Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Tangent Bundles and Vector Bundles Vector Fields and Flows