Affine focal set

This means that the derivative of an affine principal curvature in its own affine principal direction is zero. We can use the standard ideas in singularity theory to classify, up to local diffeomorphism, the affine focal set. If the family of affine distance functions can be shown to be a certain kind of family then the local structure is known. We want the family of affine distance functions to be a versal unfolding of the singularities which arise. The affine focal set of a plane curve will generically consist of smooth pieces of curve and ordinary cusp points (semi-cubical parabolae).

Einstein–Cartan theory

Einstein-CartanEinstein-Cartan gravityEinstein–Cartan
., spacetime is assumed to have torsion in addition to curvature), and then the metric and torsion are varied independently. Let represent the Lagrangian density of matter and represent the Lagrangian density of the gravitational field. The Lagrangian density for the gravitational field in the Einstein–Cartan theory is proportional to the Ricci scalar: where g is the determinant of the metric tensor, and \kappa is a physical constant 8\pi G/c^4 involving the gravitational constant and the speed of light.

List of differential geometry topics

Constant curvature. taut submanifold. Uniformization theorem. Myers theorem. Gromov's compactness theorem. Gauss–Codazzi equations. Darboux frame. Hypersurface. Induced metric. Nash embedding theorem. minimal surface. Helicoid. Catenoid. Costa's minimal surface. Hsiang–Lawson's conjecture. Theorema Egregium. Gauss–Bonnet theorem. Chern–Gauss–Bonnet theorem. Chern–Weil homomorphism. Gauss map. Second fundamental form. Curvature form. Riemann curvature tensor. Geodesic curvature. Scalar curvature. Sectional curvature. Ricci curvature, Ricci flat. Ricci decomposition. Schouten tensor. Weyl curvature. Ricci flow. Einstein manifold. Holonomy. Gauss–Bonnet theorem. Hopf–Rinow theorem.

Sage Manifolds

In particular, SageManifolds implements the computation of the Riemann curvature tensor and associated objects (Ricci tensor, Weyl tensor). SageManifolds can also deal with generic affine connections, not necessarily Levi-Civita ones. More documentation is on As SageMath is, SageManifolds is a free and open source software based on the Python programming language. It is released under the GNU General Public License. To download and install SageManifolds, see here. It is more specifically GPL v2+ (meaning that a user may elect to use a licence higher than GPL version 2.) Much of the source is on tickets at

Einstein field equations

Einstein field equationEinstein's field equationsEinstein's field equation
Derivation of local energy-momentum conservation |Contracting the differential Bianchi identity with g gives, using the fact that the metric tensor is covariantly constant, i.e., The antisymmetry of the Riemann tensor allows the second term in the above expression to be rewritten: which is equivalent to using the definition of the Ricci tensor.

Saint-Venant's compatibility condition

Saint Venant compatibility equationsSaint-Venant compatibility tensor
As noted above this coincides with the vanishing of the linearization of the Riemann curvature tensor about the Euclidean metric. Saint-Venant's compatibility condition can be thought of as an analogue, for symmetric tensor fields, of Poincaré's lemma for skew-symmetric tensor fields (differential forms). The result can be generalized to higher rank symmetric tensor fields.


Low-dimensional topology is strongly geometric, as reflected in the uniformization theorem in 2 dimensions – every surface admits a constant curvature metric; geometrically, it has one of 3 possible geometries: positive curvature/spherical, zero curvature/flat, and negative curvature/hyperbolic – and the geometrization conjecture (now theorem) in 3 dimensions – every 3-manifold can be cut into pieces, each of which has one of eight possible geometries. 2-dimensional topology can be studied as complex geometry in one variable (Riemann surfaces are complex curves) – by the uniformization theorem every conformal class of metrics is equivalent to a unique complex one, and 4-dimensional topology can

Electrogravitic tensor

tidal tensor
In general relativity, the gravitoelectric tensor or tidal tensor is one of the pieces in the Bel decomposition of the Riemann tensor. It is physically interpreted as giving the tidal stresses on small bits of a material object (which may also be acted upon by other physical forces), or the tidal accelerations of a small cloud of test particles in a vacuum solution or electrovacuum solution. *Tidal tensor

Spacetime symmetries

Symmetries in General Relativitysymmetries4-dimensional spacetimes
A curvature collineation is a vector field which preserves the Riemann tensor: where R a bcd are the components of the Riemann tensor. The set of all smooth curvature collineations forms a Lie algebra under the Lie bracket operation (if the smoothness condition is dropped, the set of all curvature collineations need not form a Lie algebra). The Lie algebra is denoted by CC(M) and may be infinite-dimensional. Every affine vector field is a curvature collineation. A less well-known form of symmetry concerns vector fields that preserve the energy-momentum tensor.

Tensor contraction

For example, the Ricci tensor is a non-metric contraction of the Riemann curvature tensor, and the scalar curvature is the unique metric contraction of the Ricci tensor. One can also view contraction of a tensor field in the context of modules over an appropriate ring of functions on the manifold or the context of sheaves of modules over the structure sheaf; see the discussion at the end of this article. As an application of the contraction of a tensor field, let V be a vector field on a Riemannian manifold (for example, Euclidean space). Let be the covariant derivative of V (in some choice of coordinates).

Topogravitic tensor

In general relativity, the topogravitic tensor is one of the three pieces of the Bel decomposition of the Riemann tensor. The topogravitic tensor can be interpreted as representing the sectional curvatures for the spatial part of a frame field. * *

Sign convention

particle physicist's sign convention(+1, −1, −1, −1) signaturearbitrary
A choice of signature is associated with a variety of names: + − − −: − + + +: We catalog the choices of various authors of some graduate textbooks: : The signature + − − − corresponds to the metric tensor: : whereas the signature − + + + corresponds to: : The Ricci tensor is defined as the contraction of the Riemann tensor. Some authors use the contraction, whereas others use the alternative. Due to the symmetries of the Riemann tensor, these two definitions differ by a minus sign. In fact, the second definition of the Ricci tensor is. The sign of the Ricci tensor does not change, because the two sign conventions concern the sign of the Riemann tensor.

List of things named after Bernhard Riemann

List of topics named after Bernhard Riemannnamed after
Riemann curvature tensor also called Riemann tensor. Riemann tensor (general relativity). Pseudo-Riemannian manifold. Riemannian bundle metric. Riemannian circle. Riemannian cobordism. Riemannian connection. Riemannian connection on a surface. Riemannian cubic. Riemannian cubic polynomials. Riemannian foliation. Riemannian geometry. Fundamental theorem of Riemannian geometry. Riemannian graph. Riemannian group. Riemannian holonomy. Riemannian manifold also called Riemannian space. Riemannian metric tensor. Riemannian Penrose inequality. Riemannian polyhedron. Riemannian singular value decomposition. Riemannian submanifold. Riemannian submersion. Riemannian volume form.

Index of physics articles (R)

Riemann curvature tensor. Riemann solver. Riemann tensor (general relativity). Riemannian Penrose inequality. Riemann–Silberstein vector. Rietdijk–Putnam argument. Rietveld refinement. Riggatron. Right-hand rule. Right hand grip rule. Rigid-body kinematics. Rigid body. Rigid body dynamics. Rigid rotor. Rijke tube. Rindler coordinates. Ring-imaging Cherenkov detector. Ring current. Ring laser. Ring laser gyroscope. Ring singularity. Ring wave guide. Rip current. Ripple (electrical). Rise over thermal. Rishon model. Ritz method. Rivista del Nuovo Cimento. Roald Sagdeev. Rob Adam. Robbert Dijkgraaf. Robert A. Woodruff. Robert Adair (physicist). Robert Alfano. Robert Andrews Millikan.

Schottky problem

Riemann matricesRiemann matrixRiemann–Schottky problem
Note that a Riemann matrix is quite different from any Riemann tensor One of the major achievements of Bernhard Riemann was his theory of complex tori and theta functions. Using the Riemann theta function, necessary and sufficient conditions on a lattice were written down by Riemann for a lattice in C g to have the corresponding torus embed into complex projective space. (The interpretation may have come later, with Solomon Lefschetz, but Riemann's theory was definitive.) The data is what is now called a Riemann matrix.

Stress–energy tensor

energy–momentum tensorenergy-momentum tensorstress-energy tensor
Consequently, if \xi^{\mu} is any Killing vector field, then the conservation law associated with the symmetry generated by the Killing vector field may be expressed as : The integral form of this is : In general relativity, the symmetric stress–energy tensor acts as the source of spacetime curvature, and is the current density associated with gauge transformations of gravity which are general curvilinear coordinate transformations. (If there is torsion, then the tensor is no longer symmetric. This corresponds to the case with a nonzero spin tensor in Einstein–Cartan gravity theory.)

List of important publications in mathematics

Publications in topologyList of publications in mathematicsMémoire sur la propagation de la chaleur dans les corps solides
Groundbreaking work in differential geometry, introducing the notion of Gaussian curvature and Gauss' celebrated Theorema Egregium. * Bernhard Riemann (1854) Publication data: "Über die Hypothesen, welche der Geometrie zu Grunde Liegen", Abhandlungen der Königlichen Gesellschaft der Wissenschaften zu Göttingen, Vol. 13, 1867. English translation Riemann's famous Habiltationsvortrag, in which he introduced the notions of a manifold, Riemannian metric, and curvature tensor.

Five-dimensional space

five-dimensionalfifth dimensionfive dimensions
Minkowski space and Maxwell's equations in vacuum can be embedded in a five-dimensional Riemann curvature tensor. In 1993, the physicist Gerard 't Hooft put forward the holographic principle, which explains that the information about an extra dimension is visible as a curvature in a spacetime with one fewer dimension. For example, holograms are three-dimensional pictures placed on a two-dimensional surface, which gives the image a curvature when the observer moves. Similarly, in general relativity, the fourth dimension is manifested in observable three dimensions as the curvature path of a moving infinitesimal (test) particle.

Nordström's theory of gravitation

his theorycompeting field theoryhis previous 1912 and 1913 theories of gravitation
But taking account of the twice-contracted and detraced Bianchi identity, a differential identity which holds for the Riemann tensor in any (semi)-Riemannian manifold, we see that in Nordström's theory, as a consequence of the field equations, we have the first-order covariant differential equation :which constrains the semi-traceless part of the Riemann tensor (the one built out of the trace-free Ricci tensor). Thus, according to Nordström's theory, in a vacuum region only the semi-traceless part of the Riemann tensor can be nonvanishing.

Frobenius manifold

Frobenius manifolds
A Riemannian manifold admits a compatible affine flat structure if and only if its curvature tensor vanishes everywhere. A family of commutative products * on TM is equivalent to a section A of S 2 (T * M) ⊗ TM via We require in addition the property Therefore, the composition g # ∘A is a symmetric 3-tensor. This implies in particular that a linear Frobenius manifold (M, g, *) with constant product is a Frobenius algebra M. Given (g, T f, A), a local potential Φ is a local smooth function such that for all flat vector fields X, Y, and Z.

Gauge theory gravity

The GTG analog of the Riemann tensor is built from the commutation rules of these derivatives. : The field equations are derived by postulating the Einstein–Hilbert action governs the evolution of the gauge fields, i.e. :Minimizing variation of the action with respect to the two gauge fields results in the field equations :where \mathcal{T} is the covariant energy–momentum tensor and \mathcal{S} is the covariant spin tensor. Importantly, these equations do not give an evolving curvature of spacetime but rather merely give the evolution of the gauge fields within the flat spacetime.

Bimetric gravity

bimetric theorybimetric theories of gravitybimetric theories
Let R^{h}_{ijk} and P^{h}_{ijk} be the Riemann curvature tensors calculated from g_{ij} and \gamma_{ij}, respectively. In the above approach the curvature tensor P^{h}_{ijk} is zero, since \gamma_{ij} is the flat space-time metric. A straightforward calculation yields the Riemann curvature tensor Each term on the right hand side is a tensor. It is seen that from GR one can go to the new formulation just by replacing {:} by \Delta and ordinary differentiation by covariant \gamma-differentiation, \sqrt {-g} by, integration measure d^{4}x by, where, and. Having once introduced \gamma_{ij} into the theory, one has a great number of new tensors and scalars at one's disposal.

Codazzi tensor

Codazzi tensors (named after Delfino Codazzi) arise very naturally in the study of Riemannian manifolds with harmonic curvature or harmonic Weyl tensor. In fact, existence of Codazzi tensors impose strict conditions on the curvature tensor of the manifold. Let (M,g) be a n-dimensional Riemannian manifold for n \geq 3, let T be a tensor, and let \nabla be a Levi-Civita connection on the manifold. We say that the tensor T is a Codazzi tensor if *Weyl–Schouten theorem * Arthur Besse, Einstein Manifolds, Springer (1987).

Mathematical physics

mathematical physicistmathematicalmathematical physicists
The gravitational field is Minkowski spacetime itself, the 4D topology of Einstein aether modeled on a Lorentzian manifold that "curves" geometrically, according to the Riemann curvature tensor, in the vicinity of either mass or energy. (Under special relativity—a special case of general relativity—even massless energy exerts gravitational effect by its mass equivalence locally "curving" the geometry of the four, unified dimensions of space and time.) Another revolutionary development of the 20th century was quantum theory, which emerged from the seminal contributions of Max Planck (1856–1947) (on black-body radiation) and Einstein's work on the photoelectric effect.