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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 2001 • Question 8

Question

Math 6510 • Differentiable Manifolds Fall 2001 • Question 8

Let \(X\) be a closed manifold, \(\Delta =\{(x,x)\in X\times X\}\) the diagonal, and \[ N(\Delta ,X\times X)=\{((x,x),(v,-v))\mid x\in X,\ v\in T_x(X)\} \] the normal bundle of \(\Delta \). Show that there is a diffeomorphism of a neighborhood of \(X\) (viewed as the zero section) in the tangent bundle \(T(X)\) with a neighborhood of \(\Delta \) in \(X\times X\), extending the usual diffeomorphism \(X\to \Delta \), \(x\mapsto (x,x)\).

Raw LaTeX

Let \(X\) be a closed manifold, \(\Delta=\{(x,x)\in X\times X\}\) the diagonal, and
\[
  N(\Delta,X\times X)=\{((x,x),(v,-v))\mid x\in X,\ v\in T_x(X)\}
\]
the normal bundle of \(\Delta\). Show that there is a diffeomorphism of a neighborhood
of \(X\) (viewed as the zero section) in the tangent bundle \(T(X)\) with a neighborhood
of \(\Delta\) in \(X\times X\), extending the usual diffeomorphism
\(X\to\Delta\), \(x\mapsto(x,x)\).

Topic Tags

Diffeomorphisms and Isotopy Methods Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Tangent Bundles and Vector Bundles