# Sign convention in generalised Gauss-Bonnet

Apparently I cannot get the right sign in deriving classical Gauss-Bonnet from generalised one!

According to many references (e.g. Madsen and Tornehave, Nakahara, Milnor and Stasheff if you don't use their particular convention about orientation) generalised Gauss-Bonnet is

$\int_M Pf(-R/2\pi)=\chi(M)$

where $R$ is the curvature 2-form and the Pfaffian of a 2lx2l antisymmetric matrix is $Pf(A)=\frac{1}{2^ll!}\epsilon ^{i_1 \ldots i_{2l}}A_{i_1 i_2} \ldots A_{i_{2l-1}i_{2l}} \$ i.e. the sign convention is such that for the matrix A={{0,a},{-a,0}} Pf(A)=a.

For l=1, M is a surface and $R^a_{\phantom{a}b}=\frac{1}{2}R^a_{\phantom{a}bcd}e^c\wedge e^d =K (g_{ac}g_{bd}-g_{ad}g_{bc})$ where $K$ is the Gaussian curvature, $g_{ab}$ the metric tensor and $\{e^a\}$ an orthonormal basis. For the Pfaffian I get $Pf(-R/2 \pi)= -\frac{1}{4\pi} 2 \epsilon_{12} R^1_2= -\frac{1}{2\pi} R^1_{\phantom{1}2}=-\frac{1}{2\pi}R^1_{\phantom{1}212}e^1\wedge e^2$

$e^1\wedge e^2$ is the volume form and $R^1_{\phantom{1}212}=K$ therefore

$\int_M Pf(-R/2\pi)=\int_M-K/2\pi=\chi$

which obviously has the wrong sign! I have carefully checked the definitions in the books and I think I am using the same as they use - so please be very explicit in pointing aout where the mistake is! Thanks

• A common place to make a sign error is the curvature. Compute the curvature of a 2-sphere using your formulas and make sure it's positive. – Deane Yang Mar 22 '13 at 12:52
• It turned out that was the problem indeed: my convention for the definition of curvature 2-form where different from those used in the books, resulting in a minus sign difference. – GFR Mar 22 '13 at 15:34

## 1 Answer

Check Example 2.2.63 page 57 and pages 162-163 of these notes to see the correct sign conventions. The usual definition of the pfaffianof a $2\ell\times 2\ell$ skew-symmetric matrix is $(-1)^\ell \times$ your definition of the pfaffian; see pages 56-57 of the same notes.

• Thanks Liviu, I agree that with the conventions that you pointed out everything works out fine. I still have not accepted the answer to see if anyone can confirm that there is a mistake in the books I have mentioned or - more likely - point out what I missed out. In many sources the definition of the Pfaffian is given as $l!A\wedge \ldots \wedge A=Pf(A) vol$ which is equivalent to my definition above. – GFR Mar 22 '13 at 12:40
• Milnor an Stasheff define the pfaffian as you do but the Gauss-Bonnet formula in their case (p.311 in their book) is obtained by integrating $Pf(R/2\pi)$ and not $Pf(-R/2\pi)$ as in your case. This takes care of the the factor $(-1)^\ell$. – Liviu Nicolaescu Mar 22 '13 at 13:44
• That is true but hey also use an unconventional orientation: see page 304, note on signs. Basically if the manifold has dimension 2l and l is even their orientation is the same as the conventional one: $e^1\wedge \ldots \wedge e^{2l}$ but otherwise there is a minus sign. Quoting from the book: "Readers who prefer to use the classical sign conventions as in [Spanier], [Warner] and [Bott-Chern] can forget about these signs, but should replace K with -K whenever it occurs in our characteristic class formulas". – GFR Mar 22 '13 at 15:21
• I think I have found where the sign comes from: I was using conventions such that the connection form $\omega$ satisfies $de^i+\omega^i_{\phantom{i}j}\wedge e^j=0$, while Milnor-Stasheff has a minus sign, see pag.302. This basically correspond to replace $R^i_{\phantom{i}j}$ with $\R^j_{\phantom{i}i}=-R^i_{\phantom{i}j}$, so that an overall minus results if there is an odd number of $R^i_{\phantom{i}j}$s. – GFR Mar 22 '13 at 15:27
• In other words, Deane's comment was exactly what you needed!! – YangMills Mar 22 '13 at 15:30