Levi-Civita connection on a Riemannian surface

We are going to compute the connection form for a Riemannian surface with an orthonormal frame . And also the curvature.

Given a coframe which is orthonormal with respect to the metric $g$, we can compute the connection 1-form of the Levi-Civita connection directly from the structure coefficients of the coframe.

Suppose you have an orthonormal basis ${X_1, X_2}$ of tangent vectors (frame), and its dual coframe ${\omega^1, \omega^2}$, i.e., $\omega^i(X_j) = \delta^i_j$. The structure coefficients of the frame/coframe are defined by the commutation relations of the base vectors: $[X_i, X_j] = -T_{ij}^k X_k$. Also we have that

$d\omega^1=T_{12}^1 \omega^1 \wedge \omega^2$ and $d\omega^2=T_{12}^2 \omega^1 \wedge \omega^2$.

On the other hand, the connection 1-forms of the Levi-Civita connection, denoted by $\Theta_{j}^k$, are defined by the relation: $\nabla_{X_i} X_j = \Theta_{j}^k(X_i) X_k=\Theta_{ji}^k X_k$, where $\nabla$ is the covariant derivative associated with the Levi-Civita connection.

The Levi-Civita connection is compatible with the metric, which means that the covariant derivative of the metric is zero. In terms of the connection 1-forms, this translates into the condition: $\Theta_{ji}^k + \Theta_{ki}^j = 0$ (see here).

Now, since the connection is torsion-free

(see also here).

We can write the 8 equations:

1. $\Theta_{1}^1(X_1)-\Theta_{1}^1(X_1)=-T_{11}^1 \rightarrow 0=0$

2. $\Theta_{1}^2(X_1)-\Theta_{1}^2(X_1)=-T_{11}^2\rightarrow 0=0$

3. $\Theta_{2}^1(X_1)-\Theta_{1}^1(X_2)=-T_{12}^1$

4. $\Theta_{2}^2(X_1)-\Theta_{1}^2(X_2)=-T_{12}^2$

5. $\Theta_{1}^1(X_2)-\Theta_{2}^1(X_1)=-T_{21}^1$

6. $\Theta_{1}^2(X_2)-\Theta_{2}^2(X_1)=-T_{21}^2$

7. $\Theta_{2}^1(X_2)-\Theta_{2}^1(X_2)=-T_{22}^1\rightarrow 0=0$

8. $\Theta_{2}^2(X_2)-\Theta_{2}^2(X_2)=-T_{22}^2\rightarrow 0=0$

and we conclude:

$\Theta_{2}^1(X_1)=-T_{12}^1$ from 3,

$\Theta_{1}^2(X_2)=T_{12}^2$ from 4,

and from this two last equations:

$\Theta_{2}^1(X_2)=-\Theta_{1}^2(X_2)=-T_{12}^2$

$\Theta_{1}^2(X_1)=-\Theta_{2}^1(X_1)=T_{12}^1$

Therefore, the connection matrix is

This construction is related to Cartan's first structural equation.

Since $X_1, X_2$ is an orthonormal frame then the connection is a $\mathfrak{o}(2,\mathbb{R})$-valued 1-form $\Theta$. This is formalized by considering the associated connection in the principal bundle called orthonormal frame bundle. Or something like that...

Curvature

Since it is a particular vector bundle connection and therefore a particular connection on a fiber bundle, it has a curvature of a connection which is a 2-form of a particular kind. On the other hand, the Riemannian metric gives rise to the Riemann curvature tensor. Both ideas are related.

The curvature form, which is an $\mathfrak{o}(2,\mathbb{R})$-valued 2-form, is given by the exterior derivative $\Omega=d\Theta + \Theta \wedge \Theta=d\Theta$. To compute $d\Theta$ observe that

so

This can be renamed to

where $K$ is the Gauss curvature (see here).

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es