Q:

covariantly linear vector fields

I am looking for a generalization of linear vector fields, similar to the notion of covariantly constant vector fields:
$\nabla v=0$.
So, in cartesian coordinates (with no Christoffel symbols), we have:
$\frac{\partial v^i}{\partial x^j}=0$, that results $v^i\in \mathbb{R}$ - a constant vector
Now, take a linear vector field (in cartesian coordinates) - $v(x)=Ax$. So we have:
$\nabla v=\frac{\partial v^i}{\partial x^j}=c_k$,     for $c_k\in \mathbb{R}$
However, above equation is not invariant (since $c_k$ will transform). And I am not able to find out invariant formulation of such condition.
I will be grateful for any comments
Best Regards

A:

A linear vector field is one whose derivative is constant; so the most obvious covariant generalization is simply $\nabla \nabla v = 0.$ In the Cartesian case this reduces exactly to the second partials of $v$ being zero, which implies (on a connected domain) that $\nabla v$ is constant and thus $v$ is linear.

