Null geodesic in 2 dimensional manifold

[paweld]

I have a question. Is it true that any curve in 2-dimensional manifold which tangent vector is null at each point is null geodesic? (In 2-dimensional manifold there are only 2 null direcitions at each point).

 

[utesfan100]

For a perfect donut sitting flat on a table, the circle of contact is a curve on the torus. I may be wrong, but isn't that such a curve without being a null geodesic?

I will now eat the perfect donut.

 

[hamster143]

Only if the metric tensor is nonsingular everywhere.