How Can Reflections Change the Sign of Determinants in Manifold Atlases?

[Canavar]

Hello,

I want to show the following little exercise:

If we have a manifold M and a atlas A of M, s.t. for all coordinate maps $$x,y \in A$$:
$$det \; (d(x\circ y^{-1})<0)$$.
Then there is a atlas A' s.t. for all x',y': $$(det \; d(x\circ y'^{-1})>0)$$


I try to change the coordinate maps by reflection, i.e. if x is a coordinate map of A then take $$s \circ x$$, where s(x)=-x.

but why is then det $$d(s \circ f \circ (s \circ g)^{-1})>0$$?


Can you give me a hint?

Regards

 

[lippyka]

The key here is to note that the determinant of a matrix is unchanged when the matrix is multiplied by a scalar factor. In this case, s is a scalar factor which is -1. This means that when s is multiplied with a matrix, the determinant of the resulting matrix will be the same as the determinant of the original matrix, but with a negative sign. So, if we have a coordinate map x \in A such that det (d(x\circ y^{-1})<0), then if we take the reflection map s \circ x, the determinant of the new matrix d(s \circ x \circ (s \circ y)^{-1}) will be the same as the original matrix, but with a positive sign.