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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1994 • Question 18

Question

Math 6510 • Differentiable Manifolds Fall 1994 • Question 18

The surface parameterized by \(X:(u,v)\mapsto (\cos u,\sin u,2\cos v,2\sin v)\) is a torus in \(\mathbb {R}^4\) with the induced metric \(ds^2=X^*ds^2_{\mathbb {R}^4}\). Show that this torus cannot be isometrically immersed into \(\mathbb {R}^3\).

Raw LaTeX

The surface parameterized by
\(X:(u,v)\mapsto(\cos u,\sin u,2\cos v,2\sin v)\) is a torus in \(\mathbb{R}^4\) with
the induced metric \(ds^2=X^*ds^2_{\mathbb{R}^4}\). Show that this torus cannot be
isometrically immersed into \(\mathbb{R}^3\).

Topic Tags

Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups