- #1
orion
- 93
- 2
I don't know how to create a commutative diagram here so I'd like to refer to Diagram (1) in this Wikipedia article. I need to discuss the application of this diagram to the tangent bundle of a smooth manifold because there are some basic points that are either glossed over or conflict in the literature.
So first I'd like to examine the left hand side of that diagram and in particular ##\pi##. Some books say the following: ##\pi(p,v) = p##. Other books say ##\pi(v) = p##. Which is correct? And if it is ##\pi(p,v) = p## then how does ##\pi## differ from ##\text{proj}_1## in that diagram? And doesn't that conflict with ##\pi^{-1}(p) = T_pM## and not ##\{ p \} \times T_pM##?
I'm sorry for all the technical questions, but this is really bothering me that I can't understand that diagram as applied to the tangent bundle.
So first I'd like to examine the left hand side of that diagram and in particular ##\pi##. Some books say the following: ##\pi(p,v) = p##. Other books say ##\pi(v) = p##. Which is correct? And if it is ##\pi(p,v) = p## then how does ##\pi## differ from ##\text{proj}_1## in that diagram? And doesn't that conflict with ##\pi^{-1}(p) = T_pM## and not ##\{ p \} \times T_pM##?
I'm sorry for all the technical questions, but this is really bothering me that I can't understand that diagram as applied to the tangent bundle.