What is the defining formula for Sectional Curvature 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 http://en.wikipedia.org/wiki/Sectional_curvature
OR
$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 $\langle R(u,v)w,z \rangle = -\langle R(u,v)z,w \rangle$
 A: 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...
A: (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.
A: 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.
