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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1999 • Question 2

Question

Math 6510 • Differentiable Manifolds Fall 1999 • Question 2

Let \(M\) be a smooth manifold and \(p:E\to M\) a smooth vector bundle on \(M\) (with fibers finite-dimensional \(\mathbb {R}\)-vector spaces).

  1. Define the notion of a smooth (Euclidean, also called Riemannian) metric on \(E\).
  2. Using partitions of unity, show that \(E\) admits a smooth metric.

Raw LaTeX

Let \(M\) be a smooth manifold and \(p:E\to M\) a smooth vector bundle on \(M\) (with
fibers finite-dimensional \(\mathbb{R}\)-vector spaces).
\begin{enumerate}
\item Define the notion of a smooth (Euclidean, also called Riemannian) metric on \(E\).
\item Using partitions of unity, show that \(E\) admits a smooth metric.
\end{enumerate}

Topic Tags

Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds Tangent Bundles and Vector Bundles