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Manifold and Riemannian manifold

Shortcuts: Differences, Similarities, Jaccard Similarity Coefficient, References.

Difference between Manifold and Riemannian manifold

Manifold vs. Riemannian manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. 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.

Similarities between Manifold and Riemannian manifold

Manifold and Riemannian manifold have 30 things in common (in Unionpedia): Angle, Area, Atlas (topology), Bernhard Riemann, Carl Friedrich Gauss, Compact space, Connected space, Covering space, Curvature, Curve, Diffeomorphism, Differentiable manifold, Differential geometry, Divergence, Euclidean space, General relativity, Geodesic, Immersion (mathematics), Inner product space, Manifold, Nash embedding theorem, Pseudo-Riemannian manifold, Smoothness, Submanifold, Submersion (mathematics), Tangent space, Theorema Egregium, Torus, Vector field, Volume.

Angle

In plane geometry, an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle.

Angle and Manifold · Angle and Riemannian manifold · See more »

Area

Area is the quantity that expresses the extent of a two-dimensional figure or shape, or planar lamina, in the plane.

Area and Manifold · Area and Riemannian manifold · See more »

Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

Atlas (topology) and Manifold · Atlas (topology) and Riemannian manifold · See more »

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.

Bernhard Riemann and Manifold · Bernhard Riemann and Riemannian manifold · See more »

Carl Friedrich Gauss

Johann Carl Friedrich Gauss (Gauß; Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields, including algebra, analysis, astronomy, differential geometry, electrostatics, geodesy, geophysics, magnetic fields, matrix theory, mechanics, number theory, optics and statistics.

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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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Connected space

In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets.

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Covering space

In mathematics, more specifically algebraic topology, a covering map (also covering projection) is a continuous function p from a topological space, C, to a topological space, X, such that each point in X has an open neighbourhood evenly covered by p (as shown in the image); the precise definition is given below.

Covering space and Manifold · Covering space and Riemannian manifold · See more »

Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

Curvature and Manifold · Curvature and Riemannian manifold · 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.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

Diffeomorphism and Manifold · Diffeomorphism and Riemannian manifold · 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.

Differentiable manifold and Manifold · Differentiable manifold and Riemannian 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.

Differential geometry and Manifold · Differential geometry and Riemannian manifold · See more »

Divergence

In vector calculus, divergence is a vector operator that produces a scalar field, giving the quantity of a vector field's source at each point.

Divergence and Manifold · Divergence and Riemannian manifold · 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.

Euclidean space and Manifold · Euclidean space and Riemannian manifold · 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.

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Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

Geodesic and Manifold · Geodesic and Riemannian manifold · See more »

Immersion (mathematics)

In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.

Immersion (mathematics) and Manifold · Immersion (mathematics) and Riemannian manifold · See more »

Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

Inner product space and Manifold · Inner product space and Riemannian manifold · See more »

Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Nash embedding theorem

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space.

Manifold and Nash embedding theorem · Nash embedding theorem and Riemannian manifold · See more »

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.

Manifold and Pseudo-Riemannian manifold · Pseudo-Riemannian manifold and Riemannian manifold · See more »

Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number of derivatives it has that are continuous.

Manifold and Smoothness · Riemannian manifold and Smoothness · 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.

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Submersion (mathematics)

In mathematics, a submersion is a differentiable map between differentiable manifolds whose differential is everywhere surjective.

Manifold and Submersion (mathematics) · Riemannian manifold and Submersion (mathematics) · 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.

Manifold and Tangent space · Riemannian manifold and Tangent space · See more »

Theorema Egregium

Gauss's Theorema Egregium (Latin for "Remarkable Theorem") is a major result of differential geometry proved by Carl Friedrich Gauss that concerns the curvature of surfaces.

Manifold and Theorema Egregium · Riemannian manifold and Theorema Egregium · See more »

Torus

In geometry, a torus (plural tori) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis coplanar with the circle.

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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.

Manifold and Vector field · Riemannian manifold and Vector field · See more »

Volume

Volume is the quantity of three-dimensional space enclosed by a closed surface, for example, the space that a substance (solid, liquid, gas, or plasma) or shape occupies or contains.

Manifold and Volume · Riemannian manifold and Volume · See more »

The list above answers the following questions

Manifold and Riemannian manifold Comparison

Manifold has 286 relations, while Riemannian manifold has 73. As they have in common 30, the Jaccard index is 8.36% = 30 / (286 + 73).

References

This article shows the relationship between Manifold and Riemannian manifold. To access each article from which the information was extracted, please visit:

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