Differential Geometry: Manifolds & Curvature
Manifolds, tangent spaces, and parallel transport with curvature.
A smooth manifold $M$ of dimension $n$ is a topological space locally homeomorphic to $\mathbb R^n$, with smooth transition maps between overlapping charts. Examples: spheres $S^n$, tori, Lie groups, configuration spaces.
At each $p \in M$ sits the tangent space $T_p M$ — vectors based at $p$. A vector field is a smooth assignment $p \mapsto v_p \in T_p M$.
To compare vectors at different points, one needs a connection $\nabla$. The Levi-Civita connection (uniquely determined by a Riemannian metric $g$ via metric compatibility + torsion-freeness) defines parallel transport: sliding a vector along a curve such that $\nabla_{\dot \gamma} v = 0$.
On a curved manifold, parallel transport around a closed loop returns a different vector — the discrepancy is measured by the Riemann curvature tensor $R(X,Y)Z = \nabla_X \nabla_Y Z - \nabla_Y \nabla_X Z - \nabla_{[X,Y]} Z$. On a sphere, transporting a vector around a geodesic triangle rotates it by an angle equal to the enclosed area / $r^2$.