In Riemannian geometry, the **Rauch comparison theorem**, named after Harry Rauch who proved it in 1951, is a fundamental result which relates the sectional curvature of a Riemannian manifold to the rate at which geodesics spread apart. Intuitively, it states that for positive curvature, geodesics tend to converge, while for negative curvature, geodesics tend to spread. This theorem is formulated using Jacobi fields to measure the variation in geodesics.

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Let be Riemannian manifolds, let and be unit speed geodesic segments such that has no conjugate points along , and let be normal Jacobi fields along and such that and . Suppose that the sectional curvatures of and satisfy whenever is a 2-plane containing and is a 2-plane containing . Then for all .

In geometry, a **geodesic** is commonly a curve representing in some sense the shortest path (arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a "straight line" to a more general setting.

The **Gauss–Bonnet theorem**, or **Gauss–Bonnet formula**, is a relationship between surfaces in differential geometry. It connects the curvature of a surface to its Euler characteristic.

In differential geometry, a **Riemannian manifold** or **Riemannian space**(*M*, *g*) is a real, smooth manifold *M* equipped with a positive-definite inner product *g*_{p} on the tangent space *T*_{p}*M* at each point *p*. A common convention is to take *g* to be smooth, which means that for any smooth coordinate chart (*U*, *x*) on *M*, the *n*^{2} functions

In differential geometry, the **Ricci curvature tensor**, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space.

In Riemannian geometry, the **sectional curvature** is one of the ways to describe the curvature of Riemannian manifolds with dimension greater than 2. The sectional curvature *K*(σ_{p}) depends on a two-dimensional linear subspace σ_{p} of the tangent space at a point *p* of the manifold. It can be defined geometrically as the Gaussian curvature of the surface which has the plane σ_{p} as a tangent plane at *p*, obtained from geodesics which start at *p* in the directions of σ_{p}. The sectional curvature is a real-valued function on the 2-Grassmannian bundle over the manifold.

In Riemannian geometry, the **scalar curvature** is the simplest curvature invariant of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the intrinsic geometry of the manifold near that point. Specifically, the scalar curvature represents the amount by which the volume of a small geodesic ball in a Riemannian manifold deviates from that of the standard ball in Euclidean space. In two dimensions, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In more than two dimensions, however, the curvature of Riemannian manifolds involves more than one functionally independent quantity.

In mathematics, particularly differential geometry, a **Finsler manifold** is a differentiable manifold *M* where a **Minkowski functional***F*(*x*,−) is provided on each tangent space T_{x}*M*, that enables one to define the length of any smooth curve *γ* : [*a*,*b*] → *M* as

This is a glossary of some terms used in Riemannian geometry and metric geometry — it doesn't cover the terminology of differential topology.

In differential geometry, the **holonomy** of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geometrical data being transported. For flat connections, the associated holonomy is a type of monodromy and is an inherently global notion. For curved connections, holonomy has nontrivial local and global features.

In Riemannian geometry, a **Jacobi field** is a vector field along a geodesic in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi.

**Myers' theorem**, also known as the **Bonnet–Myers theorem**, is a celebrated, fundamental theorem in the mathematical field of Riemannian geometry. It was discovered by Sumner Byron Myers in 1941. It asserts the following:

In differential geometry, **conjugate points** or **focal points** are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are *locally* length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) *globally* length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.

In differential geometry, a subfield of mathematics, the **Margulis lemma** is a result about discrete subgroups of isometries of a non-positively curved Riemannian manifolds. Roughly, it states that within a fixed radius, usually called the **Margulis constant**, the structure of the orbits of such a group cannot be too complicated. More precisely, within this radius around a point all points in its orbit are in fact in the orbit of a nilpotent subgroup.

In mathematics, the **Abel–Jacobi map** is a construction of algebraic geometry which relates an algebraic curve to its Jacobian variety. In Riemannian geometry, it is a more general construction mapping a manifold to its Jacobi torus. The name derives from the theorem of Abel and Jacobi that two effective divisors are linearly equivalent if and only if they are indistinguishable under the Abel–Jacobi map.

In differential geometry, **Pu's inequality**, proved by Pao Ming Pu, relates the area of an arbitrary Riemannian surface homeomorphic to the real projective plane with the lengths of the closed curves contained in it.

In mathematics, a **space**, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is ; complete spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.

In mathematics, the **differential geometry of surfaces** deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric. Surfaces have been extensively studied from various perspectives: *extrinsically*, relating to their embedding in Euclidean space and *intrinsically*, reflecting their properties determined solely by the distance within the surface as measured along curves on the surface. One of the fundamental concepts investigated is the Gaussian curvature, first studied in depth by Carl Friedrich Gauss, who showed that curvature was an intrinsic property of a surface, independent of its isometric embedding in Euclidean space.

In the mathematical field of Riemannian geometry, **Toponogov's theorem** is a triangle comparison theorem. It is one of a family of theorems that quantify the assertion that a pair of geodesics emanating from a point *p* spread apart more slowly in a region of high curvature than they would in a region of low curvature.

In geometry, the **Clifton–Pohl torus** is an example of a compact Lorentzian manifold that is not geodesically complete. While every compact Riemannian manifold is also geodesically complete, this space shows that the same implication does not generalize to pseudo-Riemannian manifolds. It is named after Yeaton H. Clifton and William F. Pohl, who described it in 1962 but did not publish their result.

In mathematics, the **Cartan–Ambrose–Hicks theorem** is a theorem of Riemannian geometry, according to which the Riemannian metric is locally determined by the Riemann curvature tensor, or in other words, behavior of the curvature tensor under parallel translation determines the metric.

- do Carmo, M.P.
*Riemannian Geometry*, Birkhäuser, 1992. - Lee, J. M.,
*Riemannian Manifolds: An Introduction to Curvature*, Springer, 1997.

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