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What is the defining formula for sectional curvature?

$K_1(X,Y) = \frac{ \langle R(X,Y)Y, X \rangle} {\langle X,X \rangle \langle Y,Y \rangle - \langle X,Y \rangle} $

as in


$K_2(X,Y) = \frac{ \langle R(X,Y)X, Y \rangle} {\langle X,X \rangle \langle Y,Y \rangle - \langle X,Y \rangle} = -K_1(X,Y) $ since $\langle R(u,v)w,z \rangle = -\langle R(u,v)z,w \rangle$

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up vote 13 down vote accepted

There is no standard sign convention for the sign curvature tensor. Depending on author, the same thing is denoted as $$\langle R(X,Y)Y, X \rangle\ \ \text{or}\ \ \langle R(X,Y)X, Y \rangle.$$ But the sectional curvature is always positive for sphere and always negative for Lobachevky space. That makes you to choose one of the formulas in your question.

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But I agree completely with Anton's central point. In the end you have to decide what notation and conventions you want to use, and then the definition of sectional curvature is determined by the criteria given by Anton. – Deane Yang Apr 18 '11 at 3:17

For a discussion about the sign choice, see Lang, Fundamentals of Diff. Geom., p. 235-237. I quote him:

Classically, starting with surface theory, people wanted some formulas such as Gauss-Bonnet (...) to come out so that on the sphere, one gets a value of certain integral to be $4\pi$ and not $-4\pi$. So they picked the minus sign, and gave the notion $-R$ (normalized) the name of curvature, which makes the sphere have positive curvature.

He advocates that one choice is more natural and convenient than the other, so that the sphere ``should'' have negative curvature...

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Well, Lang is not a differential geometer, so his views are taken a lot less seriously than others. – Deane Yang Aug 27 '11 at 1:30
To be more precise, Lang is not radical: depending on what applications one makes, both R and -R are natural''. However, (...) R is the clearest functorial notion.'' To support his point, Lang says (among other things): ``The naturality of R in the real case is similar to the naturality of its counterpart in the complex case, where formulas involving positivity come out neatly by using the analogue of R rather than its negative (as already noted by Griffiths).'' Probably Griffiths is not taken seriously either... Anyway, choose your favorite definition and be happy! – Jairo Bochi Aug 27 '11 at 14:26

(This is just a comment on Anton's answer. I originally posted it as two comments, but there were too many typos.)

I agree with Anton's central point that in the end you have to decide what notation and conventions to use and the correct definition of sectional curvature is determined by the criteria given by Anton.

However, every reference I know defines $$ R(X,Y)Z = ([\nabla_X, \nabla_Y] - \nabla_{[X,Y]})Z. $$ In that case, the wikipedia definition is correct.

However, what gets confusing is that many authors want, with respect to an orthonormal frame $e_1, \dots, e_n$, the sectional curvature of the plane spanned by $e_i$ and $e_j$ to be given by $R_{ijij}$ and not $R_{ijji}$. This can be pulled off by defining $$ R^i{}_{jkl}e_i = R(e_k, e_l)e_j $$ and $R_{ijkl} = g_{ip}R^p{}_{jkl}$.

ADDED: It appears that I am wrong about the convention above being universal. However, I do believe it is the more widely used convention. In any case, Anton's advice is sound.

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Besse's book Einstein Manifolds is one that defines the curvature by putting the commutators in the other order. There are many other 'standard' examples, though you are right that the issue is further confused because one can define the positions of the raised and lowered indices however one want, and there is no consistency in how this is done. – Dan Fox Apr 18 '11 at 6:48
@Deane: There are many references which use the other convention. If you want to see something scary, look at the inside cover of Misner, Thorne and Wheeler massive treatise "Gravitation" for a list of different conventions in different GR books. And this is not just the GR community, as Dan Fox points out, the Besse collective (although perhaps not unanimously) chose the other sign for $R$. – José Figueroa-O'Farrill Apr 18 '11 at 8:11
If I remember correctly, do Carmo's "Riemannian Geometry" also uses the other convention for the sign of $R(X,Y)Z$. Personally, I use Deane's convention for $R(X,Y)Z$, but I also prefer $R_{ijkl} = R^m_{ijk} g_{ml}$ where $R^m_{ijk} e_m = R(e_i, e_j)e_k$. Since the $e_i$ and $e_j$ appear on the left of $k$ in the right hand side of the previous equation, psychologically I feel they should appear on the left of $k$ in the other side as well. And then the $(k,l)$ indices are the endomorphism of $TM$ indices, while the $(i,j)$ are the $2$-form indices, so the lowered $l$ should go right of $k$. – Spiro Karigiannis Apr 18 '11 at 11:19
All, I stand corrected. I guess I learned to ignore the conventions of certain references. – Deane Yang Apr 18 '11 at 14:43
In fact, I was taught with the second convention and find it abit more convenient, especially when dealing with calculations involving curvature as an operator. – Thomas Richard Aug 27 '11 at 10:29

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