# Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

- Second fundamental form
- In differential geometry, the second fundamental form is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by . Together with the first fundamental form, it serves to define extrinsic
- Connection form
- In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms
- Rotation number
- In mathematics, the rotation number is an invariant of homeomorphisms of the circle
- Hyperbolic set
- In dynamical systems theory, a subset Λ of a smooth manifold M is said to have a hyperbolic structure with respect to a smooth map f if its tangent bundle may be split into two invariant subbundles, one of which is contracting and the other is expanding
- Minkowski content
- The Minkowski content, or the boundary measure, of a set is a basic concept that uses concepts from geometry and measure theory to generalize the notions of length of a smooth curve in the plane, and area of a smooth surface in space, to arbitrary
- Glossary of Riemannian and metric geometry
- This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology
- Connection (mathematics)
- In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a parallel and consistent manner. There are various kinds
- Dunford–Pettis property
- In functional analysis, the Dunford–Pettis property, named after Nelson Dunford and B. J. Pettis, is a property of a Banach space stating that all weakly compact operators from this space into another Banach space are completely continuous. Many
- Sub-Riemannian manifold
- In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold. Roughly speaking, to measure distances in a sub-Riemannian manifold, you are allowed to go only along curves tangent to so-called horizontal subspaces
- Doctrine of the two swords
- In the Roman Catholicism, the doctrine of the two swords is an exegesis of Luke 22:38 elaborated in the Middle Ages. It can be understood as a particular justification for the Gelasian doctrine of "the sacred authority of the priesthood and the royal