Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1995 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 1995 • Question 8

Let \(\Delta \) be the 2-dimensional distribution in \(\mathbb {R}^3\) defined by \[ \Delta =\operatorname {span}\left \{ \frac {\partial }{\partial x^1}+x^2\frac {\partial }{\partial x^3}, \frac {\partial }{\partial x^3}+x^3\frac {\partial }{\partial x^2} \right \}. \] Determine whether this distribution is tangent to a family of two-dimensional surfaces.

Raw LaTeX

Let \(\Delta\) be the 2-dimensional distribution in \(\mathbb{R}^3\) defined
by
\[
  \Delta=\operatorname{span}\left\{
    \frac{\partial}{\partial x^1}+x^2\frac{\partial}{\partial x^3},
    \frac{\partial}{\partial x^3}+x^3\frac{\partial}{\partial x^2}
  \right\}.
\]
Determine whether this distribution is tangent to a family of two-dimensional
surfaces.

Topic Tags

Distributions, Plane Fields, and Frobenius Integrability Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups