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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2023 • Question 4

Question

Math 6510 • Differentiable Manifolds Fall 2023 • Question 4

Let \(V(x,y,z)=\frac {\partial }{\partial x}+y\frac {\partial }{\partial z}\) and \(W(x,y,z)=\frac {\partial }{\partial y}-x\frac {\partial }{\partial z}\) be vector fields on \(\mathbb {R}^3\). Show that there does not exist a surface \(S\subset \mathbb {R}^3\) such that both \(V\) and \(W\) are tangent to \(S\).

Raw LaTeX

Let \(V(x,y,z)=\frac{\partial}{\partial x}+y\frac{\partial}{\partial z}\) and
\(W(x,y,z)=\frac{\partial}{\partial y}-x\frac{\partial}{\partial z}\) be vector fields on \(\mathbb{R}^3\). Show
that there does not exist a surface \(S\subset\mathbb{R}^3\) such that both \(V\) and \(W\) are tangent to \(S\).

Topic Tags

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