Left invariant k-form

Question

I am facing a question, have just started to read a chapter from Smoth Manifolds by Lee and having trouble to do this excercise

Let w be a left invariant k-form on a Lie group G. Show that dw is a left invariant (k+1)-form.

I have re-read the definations but I can't proceed anywhere, I am new members in this site,I don't know many rules, but if I violate any, please forgive me.

Gracias señor!

Welcome to MSE. For some basic information about writing mathematics at this site see, e.g., basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. — José Carlos Santos, Mar 17 '21 at 08:19

score 0 · Accepted Answer · answered Mar 17 '21 at 09:42

0

If $\alpha$ is an invariant form and $g\in G$, denote $L_g$ the left translation, $L_g^*\alpha=\alpha$ implies that $d(L_g^*\alpha)=L_g^*d\alpha=d\alpha$ since the exterior derivative commutes with pullbacks.

Proving that the pullback map commutes with the exterior derivative

answered Mar 17 '21 at 09:42

Tsemo Aristide

89,587

Gracias Señor, I have solved it. Thank you very much again.. – Olivia Gonzalez Mar 17 '21 at 10:10

Left invariant k-form

1 Answers1