(n - 1) dimensional submanifold of the manifold R^n

Printable View

December 19th 2012, 06:15 AM
vercammen

(n - 1) dimensional submanifold of the manifold R^n

Let $A$ be a symmetric $n \times n$ matrix over $\mathbb{R}$ .
Let $0 \neq b \in \mathbb{R}$ .

Show that the surface $M = \{x\in \mathbb{R}^n \mid x^T A x = b\}$ is an $(n - 1)$ dimensional submanifold of the manifold $\mathbb{R}^n$ .

Please, help!
December 19th 2012, 06:29 AM
xxp9

Re: (n - 1) dimensional submanifold of the manifold R^n

The differential of the smooth function $f: R^n \rightarrow R, f: x \mapsto x^TAx-b$ is
$f_*: R^n \rightarrow R, f_*: v \mapsto 2x^TAv$
Since $b \ne 0$ , $x^tA$ never vanishes on the locus $f=0$ so $f_*$ has full rank.
That is, all points of M are regular points. Hence M is a n-1 dimensional sub-manifold.
December 19th 2012, 08:24 AM
vercammen

Re: (n - 1) dimensional submanifold of the manifold R^n

Thank you!!!

All times are GMT -8. The time now is 12:09 PM.

Copyright © 2005-2013 Math Help Forum. All rights reserved.

Copyright © 2005-2013 Math Help Forum. All rights reserved.