Why the multiplication map of a Lie group has constant rank?

Question

I am interested in this question. It is regarding the rank of the multiplication map of a lie group. The author states that the multiplication map has constant rank. How can I see that this is true?

Answer 1

It has constant rank because it is a submersion. It is a submersion because if you fix a coordinate, it is a diffeomorphism.

More precisely, fix $(x,y)\in G\times G$, we wish to show that $d_{(x,y)}m$ is surjective. Fix $x$ for instance, then $f:z\mapsto xz$ is a diffeomorphism (with inverse $z\mapsto x^{-1}m$), so that $d_yf : T_yG\to T_{xy}G$ is an isomorphism.

But now we have the following maps whose composite is $f$ : $G\overset{z\mapsto (x,z)}\to G\times G\overset{m}\to G$ so that $d_y f = d_{(x,y)}m\circ d_yg$ where $g:z\mapsto (x,z)$. In particular, surjectivity of $d_yf$ implies surjectivity of $d_{(x,y)}m$.

Note that, in accordance with the question you linked to, we have not used that inversion was smooth, just that $G$ was a group where the multiplication was smooth.