Let be a symmetric matrix over .

Let .

Show that the surface is an dimensional submanifold of the manifold .

Please, help!

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- Dec 19th 2012, 07:15 AMvercammen(n - 1) dimensional submanifold of the manifold R^n
Let be a symmetric matrix over .

Let .

Show that the surface is an dimensional submanifold of the manifold .

Please, help! - Dec 19th 2012, 07:29 AMxxp9Re: (n - 1) dimensional submanifold of the manifold R^n
The differential of the smooth function is

Since , never vanishes on the locus so has full rank.

That is, all points of M are regular points. Hence M is a n-1 dimensional sub-manifold. - Dec 19th 2012, 09:24 AMvercammenRe: (n - 1) dimensional submanifold of the manifold R^n
Thank you!!!