38 relations: Atlas (topology), Bilinear map, Causal structure, Causality conditions, Clifton–Pohl torus, Coordinate system, Curve, Definite quadratic form, Degenerate bilinear form, Differentiable manifold, Differential geometry, Equivalence class, Euclidean space, Fundamental theorem of Riemannian geometry, General relativity, Globally hyperbolic manifold, Hendrik Lorentz, Hopf–Rinow theorem, Hyperbolic partial differential equation, Isotropic quadratic form, Levi-Civita connection, Manifold, Metric signature, Metric tensor, Minkowski space, Orientability, Orthogonal basis, Pseudo-Euclidean space, Quadratic form, Real number, Riemann curvature tensor, Riemannian manifold, Sign convention, Spacetime, Submanifold, Tangent space, Topology, Vector space.
Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas.
New!!: Pseudo-Riemannian manifold and Atlas (topology) · See more »
Bilinear map
In mathematics, a bilinear map is a function combining elements of two vector spaces to yield an element of a third vector space, and is linear in each of its arguments.
New!!: Pseudo-Riemannian manifold and Bilinear map · See more »
Causal structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.
New!!: Pseudo-Riemannian manifold and Causal structure · See more »
Causality conditions
In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds.
New!!: Pseudo-Riemannian manifold and Causality conditions · See more »
Clifton–Pohl torus
In geometry, the Clifton–Pohl torus is an example of a compact Lorentzian manifold that is not geodesically complete.
New!!: Pseudo-Riemannian manifold and Clifton–Pohl torus · See more »
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
New!!: Pseudo-Riemannian manifold and Coordinate system · See more »
Curve
In mathematics, a curve (also called a curved line in older texts) is, generally speaking, an object similar to a line but that need not be straight.
New!!: Pseudo-Riemannian manifold and Curve · See more »
Definite quadratic form
In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.
New!!: Pseudo-Riemannian manifold and Definite quadratic form · See more »
Degenerate bilinear form
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V) given by is not an isomorphism.
New!!: Pseudo-Riemannian manifold and Degenerate bilinear form · See more »
Differentiable manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.
New!!: Pseudo-Riemannian manifold and Differentiable manifold · See more »
Differential geometry
Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.
New!!: Pseudo-Riemannian manifold and Differential geometry · See more »
Equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation) defined on them, then one may naturally split the set S into equivalence classes.
New!!: Pseudo-Riemannian manifold and Equivalence class · See more »
Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
New!!: Pseudo-Riemannian manifold and Euclidean space · See more »
Fundamental theorem of Riemannian geometry
In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.
New!!: Pseudo-Riemannian manifold and Fundamental theorem of Riemannian geometry · See more »
General relativity
General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
New!!: Pseudo-Riemannian manifold and General relativity · See more »
Globally hyperbolic manifold
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold).
New!!: Pseudo-Riemannian manifold and Globally hyperbolic manifold · See more »
Hendrik Lorentz
Hendrik Antoon Lorentz (18 July 1853 – 4 February 1928) was a Dutch physicist who shared the 1902 Nobel Prize in Physics with Pieter Zeeman for the discovery and theoretical explanation of the Zeeman effect.
New!!: Pseudo-Riemannian manifold and Hendrik Lorentz · See more »
Hopf–Rinow theorem
Hopf–Rinow theorem is a set of statements about the geodesic completeness of Riemannian manifolds.
New!!: Pseudo-Riemannian manifold and Hopf–Rinow theorem · See more »
Hyperbolic partial differential equation
In mathematics, a hyperbolic partial differential equation of order n is a partial differential equation (PDE) that, roughly speaking, has a well-posed initial value problem for the first derivatives.
New!!: Pseudo-Riemannian manifold and Hyperbolic partial differential equation · See more »
Isotropic quadratic form
In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero.
New!!: Pseudo-Riemannian manifold and Isotropic quadratic form · See more »
Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.
New!!: Pseudo-Riemannian manifold and Levi-Civita connection · See more »
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
New!!: Pseudo-Riemannian manifold and Manifold · See more »
Metric signature
The signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis.
New!!: Pseudo-Riemannian manifold and Metric signature · See more »
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.
New!!: Pseudo-Riemannian manifold and Metric tensor · See more »
Minkowski space
In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.
New!!: Pseudo-Riemannian manifold and Minkowski space · See more »
Orientability
In mathematics, orientability is a property of surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector at every point.
New!!: Pseudo-Riemannian manifold and Orientability · See more »
Orthogonal basis
In mathematics, particularly linear algebra, an orthogonal basis for an inner product space is a basis for whose vectors are mutually orthogonal.
New!!: Pseudo-Riemannian manifold and Orthogonal basis · See more »
Pseudo-Euclidean space
In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional ''n''-space together with a non-degenerate quadratic form.
New!!: Pseudo-Riemannian manifold and Pseudo-Euclidean space · See more »
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.
New!!: Pseudo-Riemannian manifold and Quadratic form · See more »
Real number
In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.
New!!: Pseudo-Riemannian manifold and Real number · See more »
Riemann curvature tensor
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.
New!!: Pseudo-Riemannian manifold and Riemann curvature tensor · See more »
Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
New!!: Pseudo-Riemannian manifold and Riemannian manifold · See more »
Sign convention
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary.
New!!: Pseudo-Riemannian manifold and Sign convention · See more »
Spacetime
In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.
New!!: Pseudo-Riemannian manifold and Spacetime · See more »
Submanifold
In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.
New!!: Pseudo-Riemannian manifold and Submanifold · See more »
Tangent space
In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.
New!!: Pseudo-Riemannian manifold and Tangent space · See more »
Topology
In mathematics, topology (from the Greek τόπος, place, and λόγος, study) is concerned with the properties of space that are preserved under continuous deformations, such as stretching, crumpling and bending, but not tearing or gluing.
New!!: Pseudo-Riemannian manifold and Topology · See more »
Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
New!!: Pseudo-Riemannian manifold and Vector space · See more »
Redirects here:
Lorentz manifold, Lorentz metric, Lorentzian manifold, Lorentzian manifolds, Lorentzian metric, Pseudo Riemannian manifold, Pseudo Riemannian metric, Pseudo-Riemannian, Pseudo-Riemannian geometry, Pseudo-Riemannian metric, Pseudo-Riemannian space, Pseudo-riemannian manifold, Pseudo-riemannian metric, Pseudoriemannian manifold, Pseudoriemannian metric, Semi-Riemannian geometry, Semi-Riemannian manifold.
References
[1] https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold