Timeline for Riemannian vector bundle [closed]
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Dec 14, 2021 at 12:54 | history | closed |
abx Ben McKay Daniele Tampieri Igor Belegradek Bugs Bunny |
Not suitable for this site | |
| Dec 12, 2021 at 16:51 | vote | accept | chan | ||
| Dec 11, 2021 at 8:18 | history | edited | YCor | CC BY-SA 4.0 |
removed capitals from title
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| Dec 11, 2021 at 5:41 | review | Close votes | |||
| Dec 14, 2021 at 12:56 | |||||
| Dec 11, 2021 at 5:30 | comment | added | chan | I just started to read the book, and already like his explanation style. Thank you for the answer, it answers some other questions I had too. | |
| Dec 11, 2021 at 4:26 | answer | added | Quarto Bendir | timeline score: 2 | |
| Dec 11, 2021 at 4:13 | comment | added | Quarto Bendir | Your formula cannot be correct since the RHS is tensorial in X,Y and the LHS is not. Metric compatibility of connection D means X<s,t> = <D_Xs,t> + <s,D_Xt>. Under this condition one has the skew-symmetry <R(X,Y)s,t>+<s,R(X,Y)t> = 0. Under the rank-one condition, this skew-symmetry implies R(X,Y)=0. (When I was learning this material the most useful textbook I found was Morita "Geometry of Differential Forms.") | |
| Dec 11, 2021 at 3:53 | comment | added | chan | I'm not familiar with metric compatibility on bundles. Is what you said translate to [X,Y] <s_i,s_j> = <R(X,Y)s_i, s_j> + <s_i, R(X,Y)s_j> = 0? Here s_i, s_j are smooth sections and X,Y are tangent vectors. R is the curvature. | |
| Dec 11, 2021 at 3:17 | comment | added | Quarto Bendir | 1) You can put a bundle metric on any vector bundle using a partition of unity. 2) The curvature of a metric-compatible connection is skew-symmetric and a skew-symmetric map of a one-dimensional space is zero. | |
| S Dec 11, 2021 at 2:26 | review | First questions | |||
| Dec 11, 2021 at 5:31 | |||||
| S Dec 11, 2021 at 2:26 | history | asked | chan | CC BY-SA 4.0 |