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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2021 • Question 5

Question

Math 6510 • Differentiable Manifolds Spring 2021 • Question 5

Define two vector fields \(V\) and \(W\) on \(\mathbb {R}^3\) as follows: at \(p=(a,b,c)\) let

\[ V(p)=\frac {\partial }{\partial x}+b\frac {\partial }{\partial z} \quad \text {and}\quad W(p)=\frac {\partial }{\partial y}. \]

Prove that there is no nonempty surface \(S\subset \mathbb {R}^3\) such that both \(V\) and \(W\) are tangent to \(S\) at each of its points.

Raw LaTeX

Define two vector fields \(V\) and \(W\) on \(\mathbb{R}^3\) as follows: at \(p=(a,b,c)\) let

\[
V(p)=\frac{\partial}{\partial x}+b\frac{\partial}{\partial z}
\quad\text{and}\quad
W(p)=\frac{\partial}{\partial y}.
\]
Prove that there is no nonempty
surface \(S\subset\mathbb{R}^3\) such that both \(V\) and \(W\) are tangent to \(S\) at each of
its points.

Topic Tags

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