I'm currently an advanced undergrad studentwith interest in differential geometry and topology, but this term I took a lie groups class and it was fascinating, nevertheless, there seems to be a break in the theory at some point: You either study the algebra o the group of you study the geometry of the group.
Looking for further reading material I came across Duistermaat's book on Lie groups and it has alot of deatiled calculation that other books do not have (such as seeing the flow in the fréchet sense and proving conditions for those terribly abstract differential equations)but there seems to be a gap. That is certainly not an introductory book or even for a second course but more as an advanced book on the geometry of lie groups.
So, are there any other nice references that deal with the geometric part of lie groups?
(I am particualry interested in looking for a roadmap that let's me deal with geometric flows on lie groups such as ricci flow, mean curvature flow and harmonic functional on lie groups such as the area functional, and so on)
Thank you all for your time reading this.
It might be not what you want.
I think it's a dilemma: You want to study the geometry of lie group by staying a bit away from the algebra, but the geometry is interesting precisely because the algebra helps simplify the geometric problem/calculations.
My suggestion is that you either take a standard course on Lie group or read a standard (not too long) book on Lie group (such as this and this).
For the geometric side, it depends on which flows you are working on. You might follow the suggestions here for Ricci flows and mean curvature flows.