Timeline for Classification of natural invariants of Riemannian structures
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
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| Apr 5, 2015 at 10:56 | comment | added | Robert Bryant | @MKO: This isn't true already in dimension $n=3$: There, you would only get $H_0=1$ and $H_2$, the scalar curvature. However, there are two more that are algebraically independent: When $n=3$, the Riemann curvature tensor is essentially the same as the Ricci curvature, which is a quadratic form, so each of the three symmetric functions of the eigenvalues of the Ricci curvature (with respect to $g$) gives an example, with $H_2$ being the trace. When $n>2$, the number $N$ of algebraically independent rational invariants of order at most $2$ is $$N=\frac{(n{+}3)(n)(n{-}1)(n{-}2)}{12}\ .$$ | |
| Apr 5, 2015 at 3:59 | comment | added | asv | Thank you. The result which I would like to have in the case of 2 derivatives is that any such function is a polynomial in $H_e$'s (in the 2 dimensional case this would be equivalent to what you said). Also probably I do not know which Weyl's theorem you referred to. | |
| Apr 4, 2015 at 19:47 | comment | added | Robert Bryant | For surfaces, expression in $g$ that involve at most $2$ derivatives have to belong to $I_2 = \mathbb{R}[K]$, so they are only the powers of $K$ (if they have a fixed weight). This is completely classical. In higher dimensions, if it only involves derivatives of order at most $2$ in $g$, it must be a polynomial in the Riemann curvature tensor (with coefficients that could depend on $g$). Again, this is a consequence of Weyl's theorem and invariant theory. | |
| Apr 4, 2015 at 17:22 | comment | added | asv | Thanks for the interesting information. I would appreciate a reference. Also the case which might be of particular interest for me is expressions in $g$ which involve at most 2 derivatives. | |
| Apr 4, 2015 at 12:05 | history | edited | Robert Bryant | CC BY-SA 3.0 |
fixed some typos and added one more sentence about the structure of I_4
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| Apr 4, 2015 at 11:47 | history | answered | Robert Bryant | CC BY-SA 3.0 |