I understood your explanations completely. Thank you so much for the detailed reply.

Given a fixed point on a manifold M, there are two metric structures in play: the metric structure on the tangent space given by the bilinear form (which, when viewed in an orthonormal frame, is isometric to Euclidean space); and the metric structure on the manifold itself, given by the metric d defined in (7). The exponential map preserves distances between pairs of points only along rays from the origin to the cut locus. Thus, when v, w lie along a ray from the origin (and lie inside the cut locus), but in other situations one does not expect this equality to hold (although comparison geometry can yield some weak substitutes for this equality).

In general, I do not think that one can determine when two points on the cut locus map to the same point under the exponential map without using some global information about the manifold; just knowing the metric inside the cut locus is not going to be enough (there may be multiple ways to glue the boundary of this locus to itself). On the other hand, the exponential map is always injective in the interior region bounded by the cut locus.

In the case of a unit sphere, the cut locus bounds a disk of radius in the tangent plane, and the exponential map maps the entire boundary of this disk (i.e. a circle of radius ) to the antipodal point on the sphere. So there is no unique “tangent cut point” in this case - the entire cut locus is being mapped to the antipodal point.

It may be best to work through a textbook in Riemannian geometry (especially one with exercises) in order to have a firm grasp of these concepts. (A good textbook will also have some useful diagrams which should be helpful for visualising what is going on; this is hard to convey in this blog medium.)

More basic questions:

Take a sphere S^2, pick a North Pole and get its tangent plane. Again, I will call the North Pole in the tangent plane the tangent point. The cut locus in the manifold is the antipodal point (South Pole) and the set of midpoints is the equatorial sphere. Now where is the tangent cut point? It’s right “beneath” the tangent point, is it not???

Continuing the case of S^2 above, around the tangent point in the tangent plane, I see a circle of radius 1 which is a union of infinitely many endpoints of vectors emanating from the tangent point. Is this circle the tangent cut locus or the equatorial sphere when taken to the manifold through Exponential map???

Thank you.

Some basic questions about the notions of geodesic and cut locus.

I’ve read that the notion of distance on a manifold is measured in the tangent plane. Hence, does d(x_0, x) measured via (7) imply that measured in the tangent plane?

Call a point x and its cut point Cut(x) on a manifold. Call exp^(-1)(x) a tangent point and exp^(-1)(Cut(x)) a tangent cut point. Is the distance from a point to a cut point same as the distance from the tangent point to the tangent cut point?

In addition to above, denote the midpoint between x and Cut(x) by y. Is the distance between x and y same as the distance between the tangent point and a tangent cut point divided by 2?

Is Exp map restricted to tangent cut points bijective??? It is clearly surjective but how can one determine if it is also injective? In other words, how would one know if four tangent cut points all go to four different cut points or possibly all to a same cut point??? This last one has been in particular mystery to me.