Structure constants to Christoffel Symbols

Question

What is meant when one says that one has chosen a basis of fields on the manifold with ``anolonomy"?

I get the feeling that it is a choice of basis with non-trivial structure constants say $C^{k}_{ij}$

Now in some papers people seem to state that once such a basis is chosen one can write the Christoffel symbols in terms of the structure constants by the following equation,

I am writing the equation with all indices lowered (as written in the original paper)

$\Gamma_{ijk} = \frac{1}{2}(C_{ijk} - C_{ikj} - C_{jki})$

I suppose they are assuming that the connection is chosen to be torsion free since the above equation satisfies torsion free-ness condition.

But just being torsion free doesn't seem to be enough to derive the above relationship.

This relationship is seen in the papers in the context of choosing a veirbein on homogeneous spaces equipped with a Riemannian metric.

I would like to know from where and how does the above equation come.

In general it is surely impossible that the structure constants determine the connection!

Answer 1

Maybe I should elaborate on my comment. On a riemannian manifold $(M,g)$ there exists a unique metric-compatible torsion-free affine connection. It's the Levi-Civita connection and one can prove its existence and uniqueness constructively by giving a formula, known as the Koszul formula. This formula is given by $$2 g(\nabla_XY,Z) = X g(Y,Z) + Y g(Z,X) - Z g(X,Y) - g(Y,[X,Z]) - g([Y,Z],X) - g(Z,[X,Y]) $$ where $X,Y,Z$ are vector fields on $M$. Now take a local orthonormal frame $(e_i)$ for the tangent bundle. Orthonormality says that $g(e_i,e_j)$ is constant, whereas because they are a frame $[e_i,e_j] = C_{ij}^k e_k$ for some functions $C_{ij}^k$. (They will not be constant in general.) If you now apply the Koszul formula to the elements in this frame you get your expression, where $$C_{ijk} = g([e_i,e_j],e_k).$$

Answer 2

You are correct, a nonholonomic (or anholonomic) basis of the tangent bundle is a set of vector fields $X_i$, $i = 1, \ldots, n$, with nonvanishing structure constants: $[X_i, X_j] = C_{ij}^k X_k$. I'm not sure where this terminology comes from, since in classical mechanics nonholonomicity means exactly the opposite (i.e. the Lie bracket is not closed).

I haven't seen the formula you wrote down. The indices seem to be in the wrong places in the second and third term, could you check this?

Not sure if this helps, but on a Lie group at least there is a connection $\nabla$ defined in terms of the Lie bracket of left invariant vector fields as

$\nabla_X Y = \frac{1}{2} [X, Y]$

So the Christoffel symbols for this metric are just the structure constants of the Lie group. This is due to Milnor.

I guess this could be extended to the case of homogeneous manifolds, but at some stage something has got to give since the Lie bracket is not a connection (not tensorial in the first argument). So in that case, maybe your term 2 and 3 are the necessary correction terms.