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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Winter 2004 • Question 3

Question

Math 6510 • Differentiable Manifolds Winter 2004 • Question 3

Consider vector fields in \(\mathbb {R}^3\): \[ X=x\frac {\partial }{\partial x}+\frac {\partial }{\partial y} \qquad Y=\frac {\partial }{\partial x}-\frac {\partial }{\partial z}. \] Show that there is no nonempty surface \(S\subset \mathbb {R}^3\) that is tangent to both vector fields at each of its points.

Raw LaTeX

Consider vector fields in \(\mathbb{R}^3\):
\[
  X=x\frac{\partial}{\partial x}+\frac{\partial}{\partial y}
  \qquad
  Y=\frac{\partial}{\partial x}-\frac{\partial}{\partial z}.
\]
Show that there is no nonempty surface \(S\subset\mathbb{R}^3\) that is tangent to both
vector fields at each of its points.

Topic Tags

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