Assume that (M,g) is a Riemannian manifold with the following property:
For every p in M, all non singular trajectories of the gradient vector field correspond to function d^2(x,p) are geodesics(after a possible reparametrization).Here d is the metric arising from the Riemannian metric.
To what extent such Riemannian manifolds are studied?
An obvious example is the standard Riemanian structure of the Euclidian space.
What are some non trivial examples?Are there examples among Riemann surfaces?
On the other side of the question, do you have any Riemannian manifold for which your conjecture does not hold?
Dear Prof. Leyvraz Thank you for your answer After your answer, I search more and I understand from proposition 1.8 of this file that the solutions are always geodesics(At least locally).
On the other hand it is difficult to imagine this fact for the torus, since the gradient vector field has singularities of negative index, say saddle point. It is difficult to imagine that the geodesics around a saddle point are so divergent from each other
http://cedricvillani.org/wp-content/uploads/2012/08/P12.CIME_.pdf
The distance d^2(x,p) is ``multivalued'' in the large on manifolds like a torus, so any arguments in the large are tricky.
This property is true for any Riemannian manifold. If you use the function d instead of d^2, the trajectories of the gradient (whereever d is smooth) are geodesics parametrized by arc length.
Best regards
Jost
Hi, If \nabla denotes the Levi-Civita connection on he Riemannian manifold (M,g) and D/dt denotes the related covariantt derivative, and the trajectories x(t) of the function d^2(x,p) are geodesics (after a reparametrization) then in any local chart on M, (D/dt v(t)) is proportional to v(t), where, v(t)=d/dt(d(x(t)). Moreover, the function f(x):=d^2(x,p) is a solution to the following PDE
\nabla_i \nabla_j f=\lambda(f) g_ij,
where, \lambda is an appropriate function defined on M and also is a function of f. This manifolds are studied upon special forms of \lambda (e.g polynomial on f). See for example
Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc., 117 (1965), pp. 251–275.
The above study may help you to discover the Riemannian manifold satisfying your property (f may be replaced by d^2(x,p)).
All the best
I woild like to sincerely than all of you for your interesting answers and valuable references.
The main motivation for this question was my following MO conversation
http://mathoverflow.net/questions/261627/a-model-of-smooth-projective-plane
Best regards
Ali Taghavi
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