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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 9

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 9

Suppose a smooth oriented manifold \(M^n\) has a Riemannian metric which is given in any local coordinate system by \[ ds^2=\sum _{i,j=1}^{n}g_{ij}(x)\,dx^i\otimes dx^j. \] Then the form given locally in the same coordinate systems by \[ \omega =\det (g_{ij}(x))\,dx^1\cdots dx^n \] represents a globally defined nowhere vanishing \(n\)-form on \(M\).

Raw LaTeX

Suppose a smooth oriented manifold \(M^n\) has a Riemannian metric
which is given in any local coordinate system by
\[
  ds^2=\sum_{i,j=1}^{n}g_{ij}(x)\,dx^i\otimes dx^j.
\]
Then the form given locally in the same coordinate systems by
\[
  \omega=\det(g_{ij}(x))\,dx^1\cdots dx^n
\]
represents a globally defined nowhere vanishing \(n\)-form on \(M\).

Topic Tags

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