A curve $\alpha$ on a riemannian manifold $(M,g,\nabla)$ is a geodesic if $\nabla_TT=0$, where $T$ is the tangent vector field. A generalization of this geodesic equation suggests that $\nabla_TT=\rho T$ where $\rho$ is a real valued function. To get such curves one can define a new symmetric non-metric connection
$$\bar \nabla_XY=\nabla_XY+u(X)Y+u(Y)X$$
where $u$ is a 1-form. In this case the geodesics of the new connection satisfies
$$\nabla_TT=-2u(T)T$$
If we could determine a 1-form $u$ such that $-2u(T)=\rho$, then the geodesics of the new connection are those curves that satisfy the general geodesic equation of the old connection. I have derived the curvature tensor of the new connection in terms of the old curvature tensor, the 2-form $du(X,Y)$ and $S(X,Y)=X(u(Y))-u(\nabla_XY)-u(X)u(Y)$. I try to get simple examples of such curves. Now I asked the following.
My question is
(1) Are there any hints to determine the 1-form using the only given condition
(2) How can I determine the importance of these curves(minimizing distance, ...).
Alex and Benoit said that $\rho$ should be zero and so my trial to fix the problem failed. But we have two important issues
(1) A new symmetric non-metric connection(I noticed that Pandey, Tripathe and Agashe defined and studied many semi-symmetric non-metric connections but I didn't realize the importance of introducing these new connections to a manifold equipped with the Levi-Civita connection) Is it important to study such new connections.
(2) General geodesics: I solved the problem for general geodesics of the plane with constant $\rho$ without using the one form, they are still straight lines with some strange properties(given initial position and initial velocity i got two straight different straight lines(a geodesic and a general geodesic) passing through the same point with the same velocity). Are there any hints to fix the problem of solving the general geodesic equation for arbitrary manifold.
Thanks in advance.
I hope the following can clarify what is discussed in the comment.
Let $\rho $ be a function on a Riemannian manifold $M$ and $\gamma:[0,1]\to M$ be a geodesic. Let $T(t)$ be the unit tangent vector. Let $U = aT$ for some positive function $a :[0,1] \to \mathbb R$. Then $\nabla_U U = \rho U $ can be written as
$\rho(\gamma(t)) a(t)T(t) = a\nabla_T (aT) = a(t) \dot a(t)T \Rightarrow \dot a(t) = \rho (\gamma(t))\Rightarrow a(t) = \int^t_0 \rho(\gamma(s))ds + a(0)$
Thus, if you consider the parametrization $c: [0,d]\to [0,1]$ such that $\dot c(s) = a(c(s))$, then the curve $\alpha = \gamma \circ c$ satisfies
$\dot \alpha (s) = a(c(s)) T(c(s)) = U(c(s))$.
Thus $\alpha$ satisfies $\nabla_{\dot \alpha} \dot \alpha = \rho \dot \alpha$. Note that $\alpha$ is geometrically the same as $\gamma$. So it makes sense that you got straight line in your calculation. That is, you got a geodesic anyway, but with some other parametrizations.