Timeline for When can a connection Induce a Riemannian metric for which it is the Levi-Civita connection?
Current License: CC BY-SA 3.0
4 events
| when toggle format | what | by | license | comment | |
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| Nov 21, 2019 at 15:28 | comment | added | Mike Cocos | It turns out that testing just one solution A will be enough. I can prove that if A* is a secondary solution and B* its square root then B( square root of A) and B* satisfy B=OB* with O an orthogonal matrix. It follows that the connection matrix obtained by using B* is skewsymmetric if and only if the connection matrix obtained from B is skewsymmetric. I don't know if it's customary to paste a full proof into a comment. I could send a pdf file if you're still interested. Thank you for all your help! | |
| Jan 12, 2018 at 1:15 | comment | added | Mike Cocos | Thank you Dr. Bryant. I am trying to understand why A has to be unique and hopefully I'll post a clarifying comment. | |
| Jan 5, 2018 at 14:26 | comment | added | Robert Bryant | Your algorithm is not complete even in dimension $2$. For example: The solution $X=A$ that one must assume to exist in Step 4 is not ever unique; one can always multiply any solution $A$ by any positive function. The question then becomes how to tell whether there is some choice of $A$ in Step 4 so that the connection forms with respect to $\sigma'$ in Step 6 are skew. Your algorithm needs to deal with testing all possible choices of solutions $A$ in order to decisively answer the question. (For example, see my remark in the penultimate paragraph of my answer about finding $f$.) | |
| Jul 22, 2016 at 22:57 | history | answered | Mike Cocos | CC BY-SA 3.0 |