the exponential map
Standing at a point on a curved space, you have a flat tangent space full of direction-and-distance instructions: 'go that way, this far'. The exponential map is the device that actually carries out such an instruction on the curved space. It takes a tangent vector and says: set off in that direction along the geodesic, walk for a distance equal to the vector's length, and report where you land. It is the bridge from the easy flat tangent space to the hard curved manifold.
Precisely, at a point p the exponential map exp_p takes a tangent vector v in T_p M and returns the point reached by travelling along the geodesic that starts at p with initial velocity v, for parameter time 1. Because a geodesic with velocity v travels a distance equal to |v|, the map turns the straight ray of length |v| in the flat tangent space into a geodesic of the same length on the manifold. Near the origin exp_p is a smooth, invertible correspondence, so its inverse gives a privileged chart called normal coordinates, in which geodesics through p look perfectly straight and the metric agrees with the flat one to first order — the cleanest possible local picture of a curved space.
The exponential map is the precise sense in which a curved space is locally like flat space: it lets you carry the orderly bookkeeping of the tangent space onto the manifold itself. But its reach is limited by curvature and by how far geodesics run before they cross or stop. On a sphere, geodesics fired in all directions from the north pole all reconvene at the south pole, so the map stops being one-to-one there — that meeting point is a conjugate point, and the radius out to which exp_p stays injective (the injectivity radius) is a basic measure of how much room the geometry gives you before it folds back on itself.
On a sphere of radius a, stand at the north pole and apply exp. A tangent vector of length L points you along a meridian and walks you a distance L. When L equals pi a (a quarter of the great-circle circumference, the distance to the equator is pi a / 2; to the south pole is pi a), every direction lands on the same south pole — so exp_p is many-to-one there, the signature of positive curvature.
exp shoots geodesics out from a point; on the sphere they all reconverge at the antipode, a conjugate point.
The exponential map is generally only a local match between tangent space and manifold; positive curvature makes geodesics refocus, so far enough out exp stops being one-to-one. The name borrows from the matrix exponential in Lie theory and has nothing to do with the function e^x in elementary calculus.