Affine connection

Question

The affine connection is not in general defined uniquely by the smooth structure and the Riemannian metric. Can you give some demonstration with some examples?

Answer 1

Take $M = \mathbb{R}^2$ with its standard metric. With respect to the standard coordinates $(x,y)$ each affine connection on $M$ is written as

$$ \nabla = \mathrm{d} + A $$

where $A$ is a 2 by 2 matrix of $1$-forms on $M$. Remarks:

• $\nabla$ is compatible with the metric if and only if $A$ is skew-symmetric, i.e. $A \in \mathfrak{o}(2)$

• $\nabla$ is the Levi-Civita connection when $A = 0$

Let $\omega$ be a $1$-form on $M$ and take

$$ A = \begin{pmatrix}0 & \omega \\ -\omega & 0 \end{pmatrix} $$

My computations lead to

$$ \nabla_X Y - \nabla_Y X = [ X,Y ] + T_\omega(X, Y) $$

where, if $X = (X^1, X^2)$ and $Y = (Y^1,Y^2)$,

$$ T_\omega(X,Y) = (\omega(X)\,Y^2 - \omega(Y)\, X^2, -\omega(X)Y^1 + \omega(Y)X^1) $$

For a suitable $\omega$ (e.g. $\omega = \mathrm{d}x$), there are vector fields $X$ and $Y$ such that $T_\omega(X,Y) \neq 0$, thus the corresponding $\nabla$ is a metric connection which is not torsionfree.