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I am preparing for a prelim. in Diff. Geometry, and here is a question

I have not been able to figure out:

I am trying to show that Sym(n) , the set of all symmetric matrices in M_n

= all nxn matrices, is a mfld. under inclusion.

I see two possible approaches here, none of which I have been able to

get an answer from:

i) Showing Sym(n) is an open subset of M_n .

This does not seem true.

every open set containing a member ( in the topology as a subset of R^(n^2))

contains both symmetric as well as non-symmetric matrices.

ii) Figuring out directly if Sym(n) is a subspace of M_n, i.e, showing that every

open set in Sym(n) equals W/\Sym(n) , with W open in M_n ( same topology

as above).

I don't know how to do this, because I cannot figure out the open sets in Sym(n).

I don't see how to use any reasonably nice map like, e.g, the Det. to express

Sym(n) as a subset of M_n. Some matrices in Sym(n) are invertible, others are

not, so I don't see how this would work. Nor do I see how to express Sym(n)

as the inverse image under a map of constant rank.

I am out of ideas. Any suggestions?.

Thanks For Any Help.

I

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# Symmetric Matrices as Submfld. of M_n. Prelim

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