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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2001 • Question 7

Question

Math 6510 • Differentiable Manifolds Fall 2001 • Question 7

Consider the vector fields \[ X=\frac {\partial }{\partial x}\qquad \text {and}\qquad Y=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z} \] on \(\mathbb {R}^3\). Prove or disprove: There is a nonempty surface \(M^2\subset \mathbb {R}^3\) such that for every \(p\in M\) the tangent space \(T_pM\) is spanned by \(X_p\) and \(Y_p\).

Raw LaTeX

Consider the vector fields
\[
  X=\frac{\partial}{\partial x}\qquad\text{and}\qquad
  Y=x\frac{\partial}{\partial y}+\frac{\partial}{\partial z}
\]
on \(\mathbb{R}^3\). Prove or disprove: There is a nonempty surface
\(M^2\subset\mathbb{R}^3\) such that for every \(p\in M\) the tangent space \(T_pM\) is
spanned by \(X_p\) and \(Y_p\).

Topic Tags

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