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85 relations: Almost flat manifold, Élie Cartan, Ball (mathematics), Busemann function, Cartan–Hadamard theorem, CAT(k) space, Cayley graph, Christoffel symbols, Collapsing manifold, Compact space, Complete metric space, Conformal map, Conjugate points, Conjugation, Connection (mathematics), Convex function, Cotangent bundle, Covariant derivative, Curvature, Curve, Cut locus, Cut locus (Riemannian manifold), Developable surface, Diffeomorphism, Differentiable manifold, Differential topology, Distance, Einstein–Cartan theory, Embedding, Euclidean space, Exponential map (Lie theory), Exponential map (Riemannian geometry), Finsler manifold, First fundamental form, Flow (mathematics), General relativity, Geodesic, Geodesic convexity, Glossary of differential geometry and topology, Glossary of topology, Gromov–Hausdorff convergence, Horosphere, Intrinsic metric, Isometry, Jacobi field, Killing vector field, Lattice (discrete subgroup), Levi-Civita connection, Lie group, Line (geometry), ..., Lipschitz continuity, List of differential geometry topics, Manifold, Mean curvature, Metric map, Metric space, Metric tensor, Minimal surface, Net (mathematics), Nilmanifold, Nilpotent group, Parallel transport, Polyhedral space, Principal curvature, Pullback, Quasi-isometry, Radius (bone), Riemann curvature tensor, Riemannian geometry, Riemannian manifold, Riemannian submersion, Second fundamental form, Semidirect product, Simplicial complex, Solvable Lie algebra, Solvmanifold, Spin–orbit interaction, Sub-Riemannian manifold, Submersion (mathematics), Systolic geometry, Tangent bundle, Torsion tensor, Trajectory, Vector field, Word metric. Expand index (35 more) »
Almost flat manifold
In mathematics, a smooth compact manifold M is called almost flat if for any \varepsilon>0 there is a Riemannian metric g_\varepsilon on M such that \mbox(M,g_\varepsilon)\le 1 and g_\varepsilon is \varepsilon-flat, i.e. for the sectional curvature of K_ we have |K_|.
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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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Ball (mathematics)
In mathematics, a ball is the space bounded by a sphere.
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Busemann function
In geometric topology, Busemann functions are used to study the large-scale geometry of geodesics in Hadamard spaces and in particular Hadamard manifolds (simply connected complete Riemannian manifolds of nonpositive curvature).
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Cartan–Hadamard theorem
In mathematics, the Cartan–Hadamard theorem is a statement in Riemannian geometry concerning the structure of complete Riemannian manifolds of non-positive sectional curvature.
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CAT(k) space
In mathematics, a \mathbf space, where k is a real number, is a specific type of metric space.
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Cayley graph
In mathematics, a Cayley graph, also known as a Cayley colour graph, Cayley diagram, group diagram, or colour group is a graph that encodes the abstract structure of a group.
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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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Collapsing manifold
In Riemannian geometry, a collapsing or collapsed manifold is an n-dimensional manifold M that admits a sequence of Riemannian metrics gi, such that as i goes to infinity the manifold is close to a k-dimensional space, where k i). The simplest example is a flat manifold, whose metric can be rescaled by 1/i, so that the manifold is close to a point, but its curvature remains 0 for all i.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complete metric space
In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary).
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Conformal map
In mathematics, a conformal map is a function that preserves angles locally.
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Conjugate points
In differential geometry, conjugate points are, roughly, points that can almost be joined by a 1-parameter family of geodesics.
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Conjugation
Conjugation or conjugate may refer to.
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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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Convex function
In mathematics, a real-valued function defined on an ''n''-dimensional interval is called convex (or convex downward or concave upward) if the line segment between any two points on the graph of the function lies above or on the graph, in a Euclidean space (or more generally a vector space) of at least two dimensions.
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Cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.
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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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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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Cut locus
The cut locus is a mathematical structure defined for a closed set S in a space X in which the length of every path is well defined.
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Cut locus (Riemannian manifold)
In Riemannian geometry, the cut locus of a point p in a manifold is roughly the set of all other points for which there are multiple minimizing geodesics connecting them from p, but it may contain additional points where the minimizing geodesic is unique, under certain circumstances.
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Developable surface
In mathematics, a developable surface (or torse: archaic) is a smooth surface with zero Gaussian curvature.
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Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.
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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 topology
In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
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Distance
Distance is a numerical measurement of how far apart objects are.
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Einstein–Cartan theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity.
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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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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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Exponential map (Lie theory)
In the theory of Lie groups, the exponential map is a map from the Lie algebra \mathfrak g of a Lie group G to the group, which allows one to recapture the local group structure from the Lie algebra.
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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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Finsler manifold
In mathematics, particularly differential geometry, a Finsler manifold is a differentiable manifold where a (possibly asymmetric) Minkowski norm is provided on each tangent space, allowing to define the length of any smooth curve as Finsler manifolds are more general than Riemannian manifolds since the tangent norms need not be induced by inner products.
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First fundamental form
In differential geometry, the first fundamental form is the inner product on the tangent space of a surface in three-dimensional Euclidean space which is induced canonically from the dot product of R3.
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Flow (mathematics)
In mathematics, a flow formalizes the idea of the motion of particles in a fluid.
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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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Geodesic convexity
In mathematics — specifically, in Riemannian geometry — geodesic convexity is a natural generalization of convexity for sets and functions to Riemannian manifolds.
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Glossary of differential geometry and topology
This is a glossary of terms specific to differential geometry and differential topology.
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Glossary of topology
This is a glossary of some terms used in the branch of mathematics known as topology.
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Gromov–Hausdorff convergence
In mathematics, Gromov–Hausdorff convergence, named after Mikhail Gromov and Felix Hausdorff, is a notion for convergence of metric spaces which is a generalization of Hausdorff convergence.
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Horosphere
In hyperbolic geometry, a horosphere (or parasphere) is a specific hypersurface in hyperbolic ''n''-space.
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Intrinsic metric
In the mathematical study of metric spaces, one can consider the arclength of paths in the space.
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Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
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Jacobi field
In Riemannian geometry, a Jacobi field is a vector field along a geodesic \gamma in a Riemannian manifold describing the difference between the geodesic and an "infinitesimally close" geodesic.
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Killing vector field
In mathematics, a Killing vector field (often just Killing field), named after Wilhelm Killing, is a vector field on a Riemannian manifold (or pseudo-Riemannian manifold) that preserves the metric.
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Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.
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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 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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Line (geometry)
The notion of line or straight line was introduced by ancient mathematicians to represent straight objects (i.e., having no curvature) with negligible width and depth.
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Lipschitz continuity
In mathematical analysis, Lipschitz continuity, named after Rudolf Lipschitz, is a strong form of uniform continuity for functions.
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List of differential geometry topics
This is a list of differential geometry topics.
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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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Mean curvature
In mathematics, the mean curvature H of a surface S is an extrinsic measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space.
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Metric map
In the mathematical theory of metric spaces, a metric map is a function between metric spaces that does not increase any distance (such functions are always continuous).
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Metric space
In mathematics, a metric space is a set for which distances between all members of the set are defined.
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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.
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Minimal surface
In mathematics, a minimal surface is a surface that locally minimizes its area.
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Net (mathematics)
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a generalization of the notion of a sequence.
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Nilmanifold
In mathematics, a nilmanifold is a differentiable manifold which has a transitive nilpotent group of diffeomorphisms acting on it.
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Nilpotent group
A nilpotent group G is a group that has an upper central series that terminates with G. Provably equivalent definitions include a group that has a central series of finite length or a lower central series that terminates with.
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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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Polyhedral space
Polyhedral space is a certain metric space.
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Principal curvature
In differential geometry, the two principal curvatures at a given point of a surface are the eigenvalues of the shape operator at the point.
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Pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fibre-product.
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Quasi-isometry
In mathematics, quasi-isometry is an equivalence relation on metric spaces that ignores their small-scale details in favor of their coarse structure.
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Radius (bone)
The radius or radial bone is one of the two large bones of the forearm, the other being the ulna.
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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.
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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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Riemannian submersion
In differential geometry, a branch of mathematics, a Riemannian submersion is a submersion from one Riemannian manifold to another that respects the metrics, meaning that it is an orthogonal projection on tangent spaces.
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Second fundamental form
In differential geometry, the second fundamental form (or shape tensor) is a quadratic form on the tangent plane of a smooth surface in the three-dimensional Euclidean space, usually denoted by \mathrm (read "two").
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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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Simplicial complex
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their ''n''-dimensional counterparts (see illustration).
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Solvable Lie algebra
In mathematics, a Lie algebra \mathfrak is solvable if its derived series terminates in the zero subalgebra.
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Solvmanifold
In mathematics, a solvmanifold is a homogeneous space of a connected solvable Lie group.
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Spin–orbit interaction
In quantum physics, the spin–orbit interaction (also called spin–orbit effect or spin–orbit coupling) is a relativistic interaction of a particle's spin with its motion inside a potential.
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Sub-Riemannian manifold
In mathematics, a sub-Riemannian manifold is a certain type of generalization of a Riemannian manifold.
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Submersion (mathematics)
In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.
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Systolic geometry
In mathematics, systolic geometry is the study of systolic invariants of manifolds and polyhedra, as initially conceived by Charles Loewner and developed by Mikhail Gromov, Michael Freedman, Peter Sarnak, Mikhail Katz, Larry Guth, and others, in its arithmetical, ergodic, and topological manifestations.
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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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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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Trajectory
A trajectory or flight path is the path that a massive object in motion follows through space as a function of time.
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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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Word metric
In group theory, a branch of mathematics, a word metric on a group G is a way to measure distance between any two elements of G. As the name suggests, the word metric is a metric on G, assigning to any two elements g, h of G a distance d(g,h) that measures how efficiently their difference g^ h can be expressed as a word whose letters come from a generating set for the group.
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References
[1] https://en.wikipedia.org/wiki/Glossary_of_Riemannian_and_metric_geometry
