Timeline for Let $G \subset \mathrm{GL}(n)$ be a matrix Lie group. Does there exist an affine connection under which the matrix and manifold exponential coincide?
Current License: CC BY-SA 4.0
7 events
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| Jan 24 at 21:55 | comment | added | user515519 | Just a simple remark to add: if a Lie group admits a bi-invariant pseudo-Riemannian metric then one-parameter groups are geodesics. In other words, the exponential map for the associated (bi-invariant) Levi-Civita connection coincides with the exponential map of the Lie group. | |
| Jan 9 at 17:37 | vote | accept | Spencer Kraisler | ||
| Jan 9 at 17:37 | comment | added | Spencer Kraisler | Wow I did not know it was this simple. Is this the Cartan connection, or is that something else? | |
| Jan 9 at 15:02 | comment | added | Mariano Suárez-Álvarez | @SpencerKraisler, just take any frame whose vector fields are left invariant, and set all the Christoffel symbols equal to zero. | |
| Jan 9 at 12:48 | history | edited | Sebastian | CC BY-SA 4.0 |
added 912 characters in body
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| Jan 9 at 5:57 | comment | added | Spencer Kraisler | Can you elaborate on this process? I do not know what the trivial connection and left trivialization mean. Could you define a frame and the resulting Christoffel symbols? I will accept for further elaboration. | |
| Jan 9 at 4:07 | history | answered | Sebastian | CC BY-SA 4.0 |