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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2022 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 2022 • Question 2

Let \(\mathcal {D}\) be the 2-dimensional distribution in \(\mathbb {R}^3\) spanned by the vector

fields \[ V=\frac {\partial }{\partial x} \quad \text {and}\quad W=x\frac {\partial }{\partial y}+\frac {\partial }{\partial z}. \] Let \(M\) be a smooth 2-manifold and \(\phi :M\to \mathbb {R}^3\) a smooth map such that \(\phi _*(p)(T_pM)\subset \mathcal {D}_{\phi (p)}\) for all \(p\in M\). Show that the derivative \(\phi _*(p):T_pM\to T_{\phi (p)}\mathbb {R}^3\) is not injective for any \(p\in M\).

Raw LaTeX

Let \(\mathcal{D}\) be the 2-dimensional distribution in \(\mathbb{R}^3\) spanned by the vector

fields
\[
V=\frac{\partial}{\partial x}
\quad\text{and}\quad
W=x\frac{\partial}{\partial y}+\frac{\partial}{\partial z}.
\]
Let \(M\) be a smooth 2-manifold and
\(\phi:M\to\mathbb{R}^3\) a smooth map such that \(\phi_*(p)(T_pM)\subset \mathcal{D}_{\phi(p)}\) for all \(p\in M\).
Show that the derivative \(\phi_*(p):T_pM\to T_{\phi(p)}\mathbb{R}^3\) is not injective for
any \(p\in M\).

Topic Tags

Distributions, Plane Fields, and Frobenius Integrability Left-Invariant Vector Fields Smooth Manifolds and Embedded Submanifolds Vector Fields and Flows