Timeline for 1-jet bundle on vector bundle with metric connection
Current License: CC BY-SA 3.0
10 events
| when toggle format | what | by | license | comment | |
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| Sep 16, 2012 at 10:52 | comment | added | Tobias Ohrmann | This is exactly what i wanted to know, sorry for covering it in a bunch of weird equations :-) And thank you for the answer! | |
| Sep 16, 2012 at 9:37 | comment | added | Michael Bächtold | That seems all right. Are you asking if sections in $E\oplus (E\otimes T^*M)$ of the special form $(\phi,\nabla\phi)$ where $\phi$ is a section $E$ are in one to one correspondence with holonomic sections of $J^1 E$? The answer is yes. | |
| Sep 16, 2012 at 7:29 | comment | added | Tobias Ohrmann | What I've done: 1. Expressing $j_x:T_x→T_{ϕ(x)}E$ in local coordinates: $j_x(∂_{x_i})=∂_{x_i}+∑^n_{j=1}y^j_i ∂_{y_j}$ 2. Splitting $j_x$ in horizontal and vertical part, according to ∇: $j_x=$ (horizontal part) $+ ∑^n_{j=1}(y^j_i−\sum^n_{k=1}Γ^j_{ik}y_k)∂_{y_j}$ 3. Project $j_x$ on the vertical part 4. Expressing $∇ϕ:T_xX→V_yY$ in local coordinates: $∇_∂_{x_i}\phi=∑^n_{j=1}(∂_{x_i}\phi_j−∑^n_{k=1}Γ^j_{ik}\phi_k)∂_{y_j} $5. Compare both expressions | |
| Sep 15, 2012 at 17:37 | comment | added | Michael Bächtold | I'm not sure I understand the question. The identification we were talking about is bijective: to every section of $J^1 E$ corresponds a section of $E\oplus(E\otimes T^*M)$. By holonomic section in $J^1 E$ you mean a section which is the prolongation of a section in $E$? | |
| Sep 15, 2012 at 14:07 | comment | added | Tobias Ohrmann | I still have one question: Does such an element $(\phi,\nabla \phi)$ exist for every $j_x=(x_i,y_j,y^j_i) \in (J^1E)_x$? As the condition, which have to be fulfilled for such an $\phi$, i get: $y_i^j-\sum_{k=1}^n \Gamma_{ik}^j y_k = \partial_{x_i}\phi^j - \sum_{k=1}^n\Gamma_{ik}^j \phi^k$ So, if $j$ is holonomic, one can choose the prolonged section of $j$ and the condition is fulfilled. Is holonomy also a necessary condition for the existence of such a section $\phi$? | |
| Sep 14, 2012 at 14:29 | vote | accept | Tobias Ohrmann | ||
| Sep 14, 2012 at 14:29 | comment | added | Tobias Ohrmann | Thank you very much, this helped me a lot, unfortunately i'm not autherized to click you answer up :) | |
| Sep 13, 2012 at 6:37 | history | edited | Michael Bächtold | CC BY-SA 3.0 |
response to comment
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| Sep 12, 2012 at 22:59 | comment | added | Tobias Ohrmann | Thank you! What i still don't understand is that an element $j\in J^1E$ builds an subspace $R \subset T_\phi E$. With a given local trivialization $\psi$, $j$ could be represented by a point in the corresponding horizontal space $HE^\psi_\phi \subset T_\phi E$ of $\psi$. Why does the set of those points in $T_\phi E$ according to the set of all trivializations build a subspace? I tried to prove that, but what I need is that the trivializations build a vector space themselves, which is obviously wrong (take [$M=\mathbb R^n, E=TM, \alpha = id =-\beta$ local trivialisations] as a counterexample). | |
| Sep 12, 2012 at 17:11 | history | answered | Michael Bächtold | CC BY-SA 3.0 |