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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 9

Let \(M^n\) be a smooth oriented manifold. Suppose \(f:M\to \mathbb {R}\) is a smooth function such that \(df\ne 0\) at each point of \(M\). Show that there is a smooth vector field \(V\) on \(M\) so that \(df(V)\le -1\). Show also, that if \(\phi _t\) is the flow generated by \(V\), then \(f(\phi _1(x))\le f(x)-1\) for all \(x\in M\) such that \(\phi _1(x)\) exists.

Raw LaTeX

Let \(M^n\) be a smooth oriented manifold. Suppose \(f:M\to\mathbb{R}\) is a smooth
function such that \(df\ne0\) at each point of \(M\). Show that there is a smooth vector
field \(V\) on \(M\) so that \(df(V)\le-1\). Show also, that if \(\phi_t\) is the flow
generated by \(V\), then \(f(\phi_1(x))\le f(x)-1\) for all \(x\in M\) such that
\(\phi_1(x)\) exists.

Topic Tags

Left-Invariant Vector Fields Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Vector Fields and Flows