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122 relations: Affine group, Affine manifold, Affine space, Affine transformation, Albert Einstein, Associated bundle, Atlas (topology), Élie Cartan, Bernhard Riemann, Bilinear map, Bundle map, Cartan connection, Cartesian product, Christoffel symbols, Complex manifold, Connection (affine bundle), Connection (fibred manifold), Connection (mathematics), Connection (principal bundle), Connection (vector bundle), Connection form, Convex set, Coordinate system, Covariant derivative, Covariant transformation, Curvature, Curve, Derivative, Development (differential geometry), Differentiable manifold, Differential (infinitesimal), Differential form, Differential geometry, Differential geometry of surfaces, Dot product, Ehresmann connection, Einstein notation, Elwin Bruno Christoffel, Embedding, Encyclopedia of Mathematics, Equivariant map, Erlangen program, Euclidean geometry, Euclidean space, Euclidean vector, Exponential map (Riemannian geometry), Exterior algebra, Exterior derivative, Felix Klein, Fiber bundle, ..., Foundations of Differential Geometry, Frame bundle, Frame of reference, Gauge covariant derivative, Gauge theory, General linear group, General relativity, Geodesic, Gregorio Ricci-Curbastro, Group action, Group homomorphism, Group representation, Hermann Weyl, Homogeneous space, Infinity, Integrability conditions for differential systems, Introduction to the mathematics of general relativity, Jean-Louis Koszul, Kernel (algebra), Klein geometry, Levi-Civita connection, Lie algebra, Lie bracket of vector fields, Lie derivative, Lie group, Linear differential equation, Linear independence, Linear map, Linearity, List of formulas in Riemannian geometry, Manifold, Mathematics, Maurer–Cartan form, Metric connection, Moving frame, Ordinary differential equation, Origin (mathematics), Overdetermined system, Parallel transport, Parallelizable manifold, Partial differential equation, Picard–Lindelöf theorem, Point (geometry), Pre-Lie algebra, Principal bundle, Principal homogeneous space, Product rule, Projective connection, Pseudo-Riemannian manifold, Pullback (differential geometry), Pullback bundle, Q.E.D., Quotient space (topology), Riemannian geometry, Riemannian manifold, Scalar (mathematics), Section (fiber bundle), Semidirect product, Smooth infinitesimal analysis, Smoothness, Tangent bundle, Tangent space, Tensor, Tensor calculus, Tensor density, Torsion tensor, Tullio Levi-Civita, Vector bundle, Vector field, Vector space, Vertical and horizontal bundles, World line. Expand index (72 more) »
Affine group
In mathematics, the affine group or general affine group of any affine space over a field K is the group of all invertible affine transformations from the space into itself.
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Affine manifold
In differential geometry, an affine manifold is a differentiable manifold equipped with a flat, torsion-free connection.
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Affine space
In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.
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Affine transformation
In geometry, an affine transformation, affine mapBerger, Marcel (1987), p. 38.
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Albert Einstein
Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).
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Associated bundle
In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with a group action of G. For a fibre bundle F with structure group G, the transition functions of the fibre (i.e., the cocycle) in an overlap of two coordinate systems Uα and Uβ are given as a G-valued function gαβ on Uα∩Uβ.
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Atlas (topology)
In mathematics, particularly topology, one describes a manifold using an atlas.
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Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Bernhard Riemann
Georg Friedrich Bernhard Riemann (17 September 1826 – 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.
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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.
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Bundle map
In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles.
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Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.
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Cartesian product
In set theory (and, usually, in other parts of mathematics), a Cartesian product is a mathematical operation that returns a set (or product set or simply product) from multiple sets.
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Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.
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Complex manifold
In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.
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Connection (affine bundle)
Let be an affine bundle modelled over a vector bundle.
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Connection (fibred manifold)
In differential geometry, a fibered manifold is surjective submersion of smooth manifolds.
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Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.
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Connection (principal bundle)
In mathematics, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points.
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Connection (vector bundle)
In mathematics, a connection on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points.
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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.
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Convex set
In convex geometry, a convex set is a subset of an affine space that is closed under convex combinations.
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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.
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Covariant derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.
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Covariant transformation
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis.
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Curvature
In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.
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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.
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Derivative
The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value).
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Development (differential geometry)
In classical differential geometry, development refers to the simple idea of rolling one smooth surface over another in Euclidean space.
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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.
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Differential (infinitesimal)
The term differential is used in calculus to refer to an infinitesimal (infinitely small) change in some varying quantity.
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Differential form
In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.
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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.
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Differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
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Dot product
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.
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Ehresmann connection
In differential geometry, an Ehresmann connection (after the French mathematician Charles Ehresmann who first formalized this concept) is a version of the notion of a connection, which makes sense on any smooth fiber bundle.
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Einstein notation
In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.
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Elwin Bruno Christoffel
Elwin Bruno Christoffel (November 10, 1829 – March 15, 1900) was a German mathematician and physicist.
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Embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
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Encyclopedia of Mathematics
The Encyclopedia of Mathematics (also EOM and formerly Encyclopaedia of Mathematics) is a large reference work in mathematics.
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Equivariant map
In mathematics, equivariance is a form of symmetry for functions from one symmetric space to another.
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Erlangen program
The Erlangen program is a method of characterizing geometries based on group theory and projective geometry.
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Euclidean geometry
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements.
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Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
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Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector (sometimes called a geometric or spatial vector, or—as here—simply a vector) is a geometric object that has magnitude (or length) and direction.
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Exponential map (Riemannian geometry)
In Riemannian geometry, an exponential map is a map from a subset of a tangent space TpM of a Riemannian manifold (or pseudo-Riemannian manifold) M to M itself.
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Exterior algebra
In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogs.
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Exterior derivative
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.
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Felix Klein
Christian Felix Klein (25 April 1849 – 22 June 1925) was a German mathematician and mathematics educator, known for his work with group theory, complex analysis, non-Euclidean geometry, and on the associations between geometry and group theory.
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Fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.
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Foundations of Differential Geometry
Foundations of Differential Geometry is an influential 2-volume mathematics book on differential geometry written by Shoshichi Kobayashi and Katsumi Nomizu.
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Frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.
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Frame of reference
In physics, a frame of reference (or reference frame) consists of an abstract coordinate system and the set of physical reference points that uniquely fix (locate and orient) the coordinate system and standardize measurements.
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Gauge covariant derivative
The gauge covariant derivative is a variation of the covariant derivative used in general relativity.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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General linear group
In mathematics, the general linear group of degree n is the set of invertible matrices, together with the operation of ordinary matrix multiplication.
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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.
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Geodesic
In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".
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Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (12January 1925) was an Italian mathematician born in Lugo di Romagna.
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Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
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Group homomorphism
In mathematics, given two groups, (G, ∗) and (H, ·), a group homomorphism from (G, ∗) to (H, ·) is a function h: G → H such that for all u and v in G it holds that where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. From this property, one can deduce that h maps the identity element eG of G to the identity element eH of H, and it also maps inverses to inverses in the sense that Hence one can say that h "is compatible with the group structure".
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Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication.
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Hermann Weyl
Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.
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Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts transitively.
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Infinity
Infinity (symbol) is a concept describing something without any bound or larger than any natural number.
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Integrability conditions for differential systems
In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms.
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Introduction to the mathematics of general relativity
The mathematics of general relativity is complex.
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Jean-Louis Koszul
Jean-Louis Koszul (January 3, 1921 – January 12, 2018) was a French mathematician, best known for studying geometry and discovering the Koszul complex.
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Kernel (algebra)
In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.
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Klein geometry
In mathematics, a Klein geometry is a type of geometry motivated by Felix Klein in his influential Erlangen program.
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Levi-Civita connection
In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie bracket of vector fields
In the mathematical field of differential topology, the Lie bracket of vector fields, also known as the Jacobi–Lie bracket or the commutator of vector fields, is an operator that assigns to any two vector fields X and Y on a smooth manifold M a third vector field denoted.
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Lie derivative
In differential geometry, the Lie derivative, named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar function, vector field and one-form), along the flow defined by another vector field.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Linear differential equation
In mathematics, a linear differential equation is a differential equation that is defined by a linear polynomial in the unknown function and its derivatives, that is an equation of the form where,..., and are arbitrary differentiable functions that do not need to be linear, and are the successive derivatives of an unknown function of the variable.
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Linear independence
In the theory of vector spaces, a set of vectors is said to be if one of the vectors in the set can be defined as a linear combination of the others; if no vector in the set can be written in this way, then the vectors are said to be.
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Linear map
In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.
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Linearity
Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line.
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List of formulas in Riemannian geometry
This is a list of formulas encountered in Riemannian geometry.
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Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Maurer–Cartan form
In mathematics, the Maurer–Cartan form for a Lie group is a distinguished differential one-form on that carries the basic infinitesimal information about the structure of.
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Metric connection
In mathematics, a metric connection is a connection in a vector bundle E equipped with a bundle metric; that is, a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve.
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Moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
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Ordinary differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation containing one or more functions of one independent variable and its derivatives.
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Origin (mathematics)
In mathematics, the origin of a Euclidean space is a special point, usually denoted by the letter O, used as a fixed point of reference for the geometry of the surrounding space.
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Overdetermined system
In mathematics, a system of equations is considered overdetermined if there are more equations than unknowns.
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Parallel transport
In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold.
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Parallelizable manifold
In mathematics, a differentiable manifold M of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at any point p of M the tangent vectors provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on M. A particular choice of such a basis of vector fields on M is called a parallelization (or an absolute parallelism) of M.
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Partial differential equation
In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.
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Picard–Lindelöf theorem
In mathematics, in the study of differential equations, the Picard–Lindelöf theorem, Picard's existence theorem or Cauchy–Lipschitz theorem is an important theorem on existence and uniqueness of solutions to first-order equations with given initial conditions.
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Point (geometry)
In modern mathematics, a point refers usually to an element of some set called a space.
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Pre-Lie algebra
In mathematics, a pre-Lie algebra is an algebraic structure on a vector space that describes some properties of objects such as rooted trees and vector fields on affine space.
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Principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group.
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Principal homogeneous space
In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial.
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Product rule
In calculus, the product rule is a formula used to find the derivatives of products of two or more functions.
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Projective connection
In differential geometry, a projective connection is a type of Cartan connection on a differentiable manifold.
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Pseudo-Riemannian manifold
In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.
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Pullback (differential geometry)
Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ*.
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Pullback bundle
In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space.
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Q.E.D.
Q.E.D. (also written QED and QED) is an initialism of the Latin phrase quod erat demonstrandum meaning "what was to be demonstrated" or "what was to be shown." Some may also use a less direct translation instead: "thus it has been demonstrated." Traditionally, the phrase is placed in its abbreviated form at the end of a mathematical proof or philosophical argument when the original proposition has been restated exactly, as the conclusion of the demonstration or completion of the proof.
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Quotient space (topology)
In topology and related areas of mathematics, a quotient space (also called an identification space) is, intuitively speaking, the result of identifying or "gluing together" certain points of a given topological space.
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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.
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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.
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Scalar (mathematics)
A scalar is an element of a field which is used to define a vector space.
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Section (fiber bundle)
In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.
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Semidirect product
In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product.
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Smooth infinitesimal analysis
Smooth infinitesimal analysis is a modern reformulation of the calculus in terms of infinitesimals.
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Smoothness
In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.
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Tangent bundle
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.
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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.
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Tensor
In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.
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Tensor calculus
In mathematics, tensor calculus or tensor analysis is an extension of vector calculus to tensor fields (tensors that may vary over a manifold, e.g. in spacetime).
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Tensor density
In differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept.
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Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.
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Tullio Levi-Civita
Tullio Levi-Civita, FRS (29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas.
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Vector bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.
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Vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.
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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.
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Vertical and horizontal bundles
In mathematics, the vertical bundle and the horizontal bundle are two subbundles of the tangent bundle of a smooth fiber bundle, forming complementary subspaces at each point of the fibre bundle.
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World line
The world line (or worldline) of an object is the path that object traces in -dimensional spacetime.
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References
[1] https://en.wikipedia.org/wiki/Affine_connection
