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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6510 • Differentiable Manifolds

Spring 2014 • Question 2

Question

Math 6510 • Differentiable Manifolds Spring 2014 • Question 2

Define \(\omega :\mathbb {R}^n\times \mathbb {R}^n\to \mathbb {R}\) by \(\omega (x,y)=x_1y_1+\cdots +x_{n-1}y_{n-1}-x_ny_n\) and define \(f:\mathbb {R}^n\to \mathbb {R}\) by \(f(x)=\omega (x,x)\). Show that \(M=f^{-1}(-1)\) is a smooth manifold. Viewing \(\omega \) as a symmetric 2-tensor on \(\mathbb {R}^n\) show that the restriction of \(\omega \) to \(M\) is positive definite. (Hint: First assume \(y\in T_xM\) and that \(x_n+y_n=0\) and show that \(\omega (x+y,x+y)=-1+\omega (y,y)\ge 0\).)

Raw LaTeX

Define \(\omega:\R^n\times\R^n\to\R\) by
\(\omega(x,y)=x_1y_1+\cdots+x_{n-1}y_{n-1}-x_ny_n\) and define \(f:\R^n\to\R\) by
\(f(x)=\omega(x,x)\). Show that \(M=f^{-1}(-1)\) is a smooth manifold. Viewing
\(\omega\) as a symmetric 2-tensor on \(\R^n\) show that the restriction of \(\omega\)
to \(M\) is positive definite. (Hint: First assume \(y\in T_xM\) and that
\(x_n+y_n=0\) and show that \(\omega(x+y,x+y)=-1+\omega(y,y)\ge0\).)

Topic Tags

Differential Forms and Exterior Calculus Homotopy and Degree on Manifolds Smooth Manifolds and Embedded Submanifolds Stokes' Theorem and Integration on Manifolds