Differential Geometry

Visualize how the Exponential Map translates flat tangent vectors into curved geodesics on manifolds.

Geodesic Equation: The "straightest" path on a curved surface with zero tangential acceleration.

$\nabla_{\dot{\gamma}} \dot{\gamma} = 0$

Exponential Map: Takes a base point $p$ and a tangent vector $v \in T_p M$, and yields the point $\gamma(1)$ reached by following the geodesic for time $t=1$.

$\exp_p(v) = \gamma(1)$

Manifold Shape

Base Point $p$

Parameter $u$0.5

Parameter $v$0.5

Tangent Vector $v \in T_p M$

Direction (Angle $\theta$)0°

Magnitude $|v|$ (Time/Length)2.0

Tangent Space (Plane)

Initial Velocity Vector ($v$)

Geodesic Trajectory

$\exp_p(v)$ (Endpoint)