Timeline for The tangent map of the exponential map
Current License: CC BY-SA 4.0
7 events
| when toggle format | what | by | license | comment | |
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| Oct 30, 2023 at 19:53 | comment | added | Jan Nienhaus | @RyanBudney This question has also been studied. The ODE is called the Jacobi equation and these fields being parallel is equivalent to the manifold in question being flat. | |
| Oct 26, 2023 at 8:18 | comment | added | Ryan Budney | I think your question boils down to the issue of if one varies the starting point of a geodesic, this process infinitesimally gives rise to a vector field along the geodesic. Your question asks if these vector fields are parallel along the geodesic. Presumably this isn't always true and there will be a relation with the Riemann curvature tensor. It might be something of the form $exp^{TM}-T(exp^M) \circ \iota$ being given in terms of some ODE involving the Riemann curvature tensor along the appropriate geodesic. | |
| Oct 26, 2023 at 4:36 | comment | added | Ryan Budney | The two maps are closer to being comparable after pre-composing one or the other with the involution of the double tangent bundle. Jan's comment is observing that the two maps take values in involuted tangent spaces. | |
| Oct 25, 2023 at 19:18 | history | became hot network question | |||
| Oct 25, 2023 at 18:13 | answer | added | Jan Nienhaus | timeline score: 9 | |
| Oct 25, 2023 at 12:28 | history | edited | YCor | CC BY-SA 4.0 |
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| Oct 25, 2023 at 5:33 | history | asked | Raz Kupferman | CC BY-SA 4.0 |