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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Fall 1998 • Question 1

Question

Math 6510 • Differentiable Manifolds Fall 1998 • Question 1

Let \(X\) be the set of points \((x,y,z)\in \mathbb {R}^3\) satisfying the equation \[ (1+z)x^2-(1-z)y^2=2z(1-z^2). \] Show that \(X\) is a smooth oriented surface embedded in \(\mathbb {R}^3\).

Raw LaTeX

Let \(X\) be the set of points \((x,y,z)\in\mathbb{R}^3\) satisfying the equation
\[
  (1+z)x^2-(1-z)y^2=2z(1-z^2).
\]
Show that \(X\) is a smooth oriented surface embedded in \(\mathbb{R}^3\).

Topic Tags

Orientability and Volume Forms Smooth Manifolds and Embedded Submanifolds Surface Topology and Surface Groups