I am a little confused about an explicit calculation of sectional curvatures. Namely, I wanted to compute the sectional curvatures of the $SU(2)$ group. The sectional curvatures of Lie groups are known to be completely described by their values in the tangent space at the identity, i.e. the Lie algebra $\mathfrak{su}(2)$ in this case. I am working with the equations in this famous paper by [Milnor][1]. There, we can see that the sectional curvatures are determined once we have given: 1) an orthonormal basis $\{e_1,e_2,\cdots,e_n\}$ for the Lie algebra, 2) the algebra itself $$[e_i,e_j]=\sum_k\alpha_{ijk}\,e_k$$ with the structure constants $\alpha_{ijk}$, and 3) a choice of inner product on the algebra $<\cdot,\cdot>$ (this is actually necessary to prove orthonormality of the basis of course). With these ingredients, the sectional curvature $\kappa(e_i,e_j)$ of the tangential 2-plane spanned by the vectors $e_i$ and $e_j$ is given by: $$\kappa(e_i,e_j)=\sum_k\left(\frac{1}{2}\alpha_{ijk}(-\alpha_{ijk}+\alpha_{jki}+\alpha_{kij})-\frac
{1}{4}(\alpha_{ijk}-\alpha_{jki}+\alpha_{kij})(\alpha_{ijk}+\alpha_{jki}-\alpha_{kij})-\alpha_{kii}\alpha_{kjj}\right)$$.
For this case, I can take the standard Pauli matrices in their Hermitian OR anti-Hermitian version (see Wikipedia for their definition) as a basis. They are both equally good orthonormal bases for the $\mathfrak{su}(2)$ algebra w.r.t. to the canonical inner product given by the scaled Killing form, $\langle A,B\rangle=\frac{1}{2}Tr(AB)$. They, however, differ in their structure constants. $\alpha_{ijk}=2i\,\epsilon_{ijk}$ for the Hermitian basis and $\alpha_{ijk}=\epsilon_{ijk}$ for the anti-Hermitian (note the extra factor of i).
My question is:
My understanding was that the sectional curvature is independent of the choice of basis in the tangent plane, and for $SU(2)$ I expect only positive sectional curvatures since it is topologically a 3-sphere (and because I am using the Killing form as an inner product). However, plugging in the above mentioned structure constants yields results with opposite signs, comming from that extra "i" in the structure constants. How can this be the case if both bases are valid?
Edit: The equation for $\kappa$ is given by Milnor only for the case $\kappa(e_1,e_2)$. I simply generalized it for all the indices, but I find it suspicious that he did not do that. Is there some detail that prevents the straighforward generalization?
[1]: https://core.ac.uk/download/pdf/82428733.pdf
Best Answer
The sectional curvature of a Riemannian manifold at a point $p$ is a function $K(X,Y)$ of couples of linearly independent vectors in $T_pM$ which happens to be dependent only on the two dimensional subspace generated by them, so it's really a function of planes in $T_pM$. In particular it's an object defined intrinsically but which, if you calculate it in a basis, depends on which basis you calculate it.
The tangent space of $SU(2)$ at the identity consists of antihermitian matrices, so the sectional curvatures computed with them is the legitimate sectional curvature.
If you calculate the sectional curvature on the hermitian matrices, that are outside the tangent space (they belong to its complexification), so that is not a sectional curvature.