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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1995 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 1995 • Question 9

Let \((M^n,g)\) be a Riemannian manifold. Show that the form given locally by \[ \omega =\sqrt {g}\,dx^1\wedge dx^2\wedge \cdots \wedge dx^n \] represents a globally-defined, nowhere vanishing \(n\)-form. (Here \(\sqrt {g}\) is the standard shorthand for the square root of the determinant of the matrix \([g_{ij}]=[\langle \frac {\partial }{\partial x^i},\frac {\partial }{\partial x^j}\rangle ]\) which gives the metric \(g\) in local coordinates.)

Raw LaTeX

Let \((M^n,g)\) be a Riemannian manifold. Show that the form given locally
by
\[
  \omega=\sqrt{g}\,dx^1\wedge dx^2\wedge\cdots\wedge dx^n
\]
represents a globally-defined, nowhere vanishing \(n\)-form. (Here \(\sqrt{g}\) is the
standard shorthand for the square root of the determinant of the matrix
\([g_{ij}]=[\langle\frac{\partial}{\partial x^i},\frac{\partial}{\partial x^j}\rangle]\)
which gives the metric \(g\) in local coordinates.)

Topic Tags

Differential Forms and Exterior Calculus Homotopy and Degree on Manifolds Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds