# Proper value function of a connection

+ 0 like - 0 dislike
176 views

Let $(E,\nabla)$ be a vector bundle with connection over $M$. I define a proper value function of $\nabla$ as a function $\lambda \in {\cal C}^{\infty}(M)$ such that:

$$\nabla_X (s)=X(\lambda).s$$

for any vector field $X$ over $M$; $s$ is a fixed section of $E$, a proper vector of the connection.

Can we decompose the space of sections of $E$ over proper vectors of the connection?

edited Mar 13, 2022

+ 0 like - 0 dislike

If $\nabla (s)=(d\lambda) s$, then we have:

$$R_{\nabla} (s)=0$$

where $R_{\nabla}$ is the curvature of the connection, so that it is maybe impossible that we had:

$$\nabla (s)= (d\lambda)s$$

answered Mar 14, 2022 by (-80 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.