Covariant derivative

covariant differentiationtensor derivativecovariant differential
This would not happen in Euclidean space and is caused by the curvature of the surface of the globe. The same effect can be noticed if we drag the vector along an infinitesimally small closed surface subsequently along two directions and then back. The infinitesimal change of the vector is a measure of the curvature.

Ricci curvature

Ricci tensorRicci curvature tensorTrace-free Ricci tensor
For any pair of tangent vectors ξ and η in T p M, the Ricci tensor Ric evaluated at is defined to be the trace of the linear map T p M → T p M given by :In local coordinates (using the Einstein summation convention), one has :where : In terms of the Riemann curvature tensor and the Christoffel symbols, one has : Due to the symmetries of the Riemann curvature tensor, it is possible for there to be a disagreement on the sign convention, since As a consequence of the Bianchi identities, the Ricci tensor of a Riemannian manifold is symmetric, in the sense that :It thus follows that the Ricci tensor is completely determined by knowing the quantity Ric for all vectors ξ of unit length.

Curvature form

Bianchi identityBianchi identitiescurvature
. * Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience. Connection (principal bundle). Basic introduction to the mathematics of curved spacetime. Chern-Simons form. Curvature of Riemannian manifolds. Gauge theory.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry.

Bernhard Riemann

RiemannGeorg Friedrich Bernhard RiemannRiemann, Bernhard
The fundamental object is called the Riemann curvature tensor. For the surface case, this can be reduced to a number (scalar), positive, negative, or zero; the non-zero and constant cases being models of the known non-Euclidean geometries. Riemann's idea was to introduce a collection of numbers at every point in space (i.e., a tensor) which would describe how much it was bent or curved. Riemann found that in four spatial dimensions, one needs a collection of ten numbers at each point to describe the properties of a manifold, no matter how distorted it is. This is the famous construction central to his geometry, known now as a Riemannian metric.

Levi-Civita connection

Christoffel symbolconnectionsLevi-Civita
Brouwer was the first mathematician to consider the parallel transport of a vector for the case of a space of constant curvature. In 1917, Levi-Civita pointed out its importance for the case of a hypersurface immersed in a Euclidean space, i.e., for the case of a Riemannian manifold embedded in a "larger" ambient space. In 1918, independently of Levi-Civita, Jan Arnoldus Schouten obtained analogous results.

Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
This allows one to speak of the Levi-Civita connection on a pseudo-Riemannian manifold along with the associated curvature tensor. On the other hand, there are many theorems in Riemannian geometry which do not hold in the generalized case. For example, it is not true that every smooth manifold admits a pseudo-Riemannian metric of a given signature; there are certain topological obstructions. Furthermore, a submanifold does not always inherit the structure of a pseudo-Riemannian manifold; for example, the metric tensor becomes zero on any light-like curve.

Jacobi field

Jacobi equation
A vector field J along a geodesic \gamma is said to be a Jacobi field if it satisfies the Jacobi equation: :where D denotes the covariant derivative with respect to the Levi-Civita connection, R the Riemann curvature tensor, the tangent vector field, and t is the parameter of the geodesic. On a complete Riemannian manifold, for any Jacobi field there is a family of geodesics \gamma_\tau describing the field (as in the preceding paragraph). The Jacobi equation is a linear, second order ordinary differential equation; in particular, values of J and at one point of \gamma uniquely determine the Jacobi field.

Christoffel symbols

Christoffel symbolChristoffel coefficientsChristoffel connection
For example, the Riemann curvature tensor can be expressed entirely in terms of the Christoffel symbols and their first partial derivatives. In general relativity, the connection plays the role of the gravitational force field with the corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γ i jk are zero. The Christoffel symbols are named for Elwin Bruno Christoffel (1829–1900).

List of formulas in Riemannian geometry

formulas in Riemannian geometryreturn to article
Ricci and scalar curvatures are contractions of the Riemann tensor. They simplify the Riemann tensor, but contain less information. The Ricci curvature tensor is essentially the unique (up to sign) nontrivial way of contracting the Riemann tensor: Due to the symmetries of the Riemann tensor, contracting on the 4th instead of the 3rd index yields the same tensor, but with the sign reversed - see sign convention (contracting on the 1st lower index results in an array of zeros). The Ricci tensor R_{ij} is symmetric.


curvednegative curvatureextrinsic curvature
See Curvature of Riemannian manifolds for the definition, which is done in terms of lengths of curves traced on the manifold, and expressed, using linear algebra, by the Riemann curvature tensor. The curvature of a differentiable curve was originally defined through osculating circles. In this setting, Augustin-Louis Cauchy showed that the center of curvature is the intersection point of two infinitely close normal lines to the curve. Intuitively, the curvature is a measure of the instantaneous rate of change of direction of a point that moves on the curve: the larger the curvature, the larger this rate of change.

Sectional curvature

curvaturecurvature tensorsmanifolds with constant sectional curvature
Riemann curvature tensor. Curvature of Riemannian manifolds. Curvature.

Scalar curvature

Ricci scalarcurvaturecurvature scalar
Among those who use index notation for tensors, it is common to use the letter R to represent three different things: These three are then distinguished from each other by their number of indices: the Riemann tensor has four indices, the Ricci tensor has two indices, and the Ricci scalar has zero indices. Those not using an index notation usually reserve R for the full Riemann curvature tensor. Alternatively, in a coordinate-free notation one may use Riem for the Riemann tensor, Ric for the Ricci tensor and R for the curvature scalar. The Yamabe problem was solved by Trudinger, Aubin, and Schoen.

Kulkarni–Nomizu product

Kulkarni-Nomizu product
The Kulkarni–Nomizu product of a pair of symmetric tensors has the algebraic symmetries of the Riemann tensor. For instance, on a 2-dimensional smooth Riemannian manifold, the Riemann curvature tensor has a simple expression in terms of the Kulkarni-Nomizu product of the metric with itself; namely, if we denote :the Riemann curvature tensor, where then where is the scalar curvature and : is the Ricci tensor, which in components reads. Proof., thus :as R_{lijm} is antisymmetric in the indices l,i; on the other hand, :since the Kulkarni-Nominzu product has the same symmetries of, as noticed before.


Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") includes the study of such topics as quantity (number theory), structure (algebra), space (geometry), and change (mathematical analysis). It has no generally accepted definition.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
.), extrinsic curvature of submanifolds, and intrinsic curvature of the manifold itself. In 1828, Carl Friedrich Gauss proved his Theorema Egregium (remarkable theorem in Latin), establishing an important property of surfaces. Informally, the theorem says that the curvature of a surface can be determined entirely by measuring distances along paths on the surface. That is, curvature does not depend on how the surface might be embedded in 3-dimensional space. See differential geometry of surfaces.

Gregorio Ricci-Curbastro

RicciGregorio Ricci CurbastroRicci-Curbastro
Ricci curvature. Ricci flow.


geodesicsgeodesic flowgeodesic equation
In a Riemannian manifold or submanifold geodesics are characterised by the property of having vanishing geodesic curvature. More generally, in the presence of an affine connection, a geodesic is defined to be a curve whose tangent vectors remain parallel if they are transported along it. Applying this to the Levi-Civita connection of a Riemannian metric recovers the previous notion. Geodesics are of particular importance in general relativity. Timelike geodesics in general relativity describe the motion of free falling test particles.

Connection form

connectionconnection one-formvertical subspace
,n}, denotes the dual basis of the cotangent bundle, such that θ i (e j ) = δ i j (the Kronecker delta), then the connection form is : In terms of the connection form, the exterior connection on a vector field v = Σ i e i v i is given by :One can recover the Levi-Civita connection, in the usual sense, from this by contracting with e i : : The curvature 2-form of the Levi-Civita connection is the matrix (Ω i j ) given by :For simplicity, suppose that the frame e is holonomic, so that dθ i =0. Then, employing now the summation convention on repeated indices, :where R is the Riemann curvature tensor.

Conformal geometry

conformal structureconformal manifoldconformal
A conformal metric is conformally flat if there is a metric representing it that is flat, in the usual sense that the Riemann curvature tensor vanishes. It may only be possible to find a metric in the conformal class that is flat in an open neighborhood of each point. When it is necessary to distinguish these cases, the latter is called locally conformally flat, although often in the literature no distinction is maintained. The n-sphere is a locally conformally flat manifold that is not globally conformally flat in this sense, whereas a Euclidean space, a torus, or any conformal manifold that is covered by an open subset of Euclidean space is (globally) conformally flat in this sense.

Constant curvature

curvatureconstantconstant curve
Euclidean geometry – constant vanishing sectional curvature. hyperbolic geometry – constant negative sectional curvature. Every space of constant curvature is locally symmetric, i.e. its curvature tensor is parallel. Every space of constant curvature is locally maximally symmetric, i.e. it has number of local isometries, where n is its dimension. Conversely, there exists a similar but stronger statement: every maximally symmetric space, i.e. a space which has (global) isometries, has constant curvature.

Exterior covariant derivative

covariant exterior derivativeexterior covariant differentiation
Identifying tensorial forms and E-valued forms, one may show that : which can be easily recognized as the definition of the Riemann curvature tensor on Riemannian manifolds. * If ω is the connection form on P, then Ω = Dω is called the curvature form of ω. Bianchi's second identity, which says that the exterior covariant derivative of Ω is zero (that is, DΩ = 0) can be stated as:.

Second fundamental form

extrinsic curvaturesecondshape tensor
In Euclidean space, the curvature tensor of a submanifold can be described by the following formula: :This is called the Gauss equation, as it may be viewed as a generalization of Gauss's Theorema Egregium. For general Riemannian manifolds one has to add the curvature of ambient space; if N is a manifold embedded in a Riemannian manifold (M,g) then the curvature tensor R N of N with induced metric can be expressed using the second fundamental form and R M, the curvature tensor of M * * Steven Verpoort (2008) Geometry of the Second Fundamental Form: Curvature Properties and Variational Aspects from Katholieke Universiteit Leuven. First fundamental form. Gaussian curvature.

Curvature tensor

Curvature tensor (disambiguation)
In differential geometry, the term curvature tensor may refer to: *Tensor (disambiguation) the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds. the curvature of an affine connection or covariant derivative (on tensors). the curvature form of an Ehresmann connection: see Ehresmann connection, connection (principal bundle) or connection (vector bundle). It is the one of the numbers that are important in the Einstein field equations.

Chern–Gauss–Bonnet theorem

Chern theoremChern-Gauss-Bonnet theoremgeneralized Gauss–Bonnet theorem
The theorem has also found numerous applications in physics, including: In dimension 2n=4, for a compact oriented manifold, we get where \text{Riem} is the full Riemann curvature tensor, \text{Ric} is the Ricci curvature tensor, and R is the scalar curvature. This is particularly important in general relativity, where spacetime is viewed as a 4-dimensional manifold. The Gauss–Bonnet theorem is a special case when M is a 2d manifold. It arises as the special case where the topological index is defined in terms of Betti numbers and the analytical index is defined in terms of the Gauss–Bonnet integrand.