Riemannian geometry

Riemannian geometry studies smooth manifolds equipped with smoothly varying inner products on tangent spaces. These metrics define lengths, angles, distances, and curvature.

Smooth manifold: A space locally modeled on Euclidean space, with smoothly compatible coordinate charts. It is the underlying space on which a Riemannian metric is defined.
Levi-Civita connection: The unique connection preserving a Riemannian metric and having zero torsion. It defines parallel transport and geodesics canonically from the metric.
General relativity: A theory of gravity in which spacetime geometry is determined by matter and energy. Its spatial geometry uses Riemannian methods, while spacetime itself has a Lorentzian metric.
Pseudo-Riemannian geometry: The study of manifolds with nondegenerate metrics of mixed signature. Unlike Riemannian metrics, its inner products need not be positive definite.
Tangent space: A vector space of directions at a point of a smooth manifold. The metric assigns an inner product to each tangent space.
Covariant derivative: A derivative operation for vector fields that accounts for changing tangent spaces. It makes differentiation of vector fields meaningful across a curved manifold.
Shape analysis: The mathematical study of shapes, their variations, and comparisons. Riemannian metrics can define distances and geodesics between shapes.
Finsler geometry: The study of manifolds whose tangent vectors have smoothly varying, generally nonquadratic norms. It generalizes length geometry beyond norms induced by inner products.
Riemannian metric: A smoothly varying positive-definite inner product on the tangent spaces of a smooth manifold. It supplies the lengths and angles that define Riemannian geometry.
Riemann curvature tensor: A tensor measuring the failure of covariant derivatives to commute. It encodes the manifold's full intrinsic curvature.

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