HOME





Conjugate Point
In differential geometry, conjugate points or focal points are, roughly, points that can almost be joined by a 1-parameter family of geodesics. For example, on a sphere, the north-pole and south-pole are connected by any meridian. Another viewpoint is that conjugate points tell when the geodesics fail to be length-minimizing. All geodesics are ''locally'' length-minimizing, but not globally. For example on a sphere, any geodesic passing through the north-pole can be extended to reach the south-pole, and hence any geodesic segment connecting the poles is not (uniquely) ''globally'' length minimizing. This tells us that any pair of antipodal points on the standard 2-sphere are conjugate points.Cheeger, Ebin. ''Comparison Theorems in Riemannian Geometry''. North-Holland Publishing Company, 1975, pp. 17-18. Definition Suppose ''p'' and ''q'' are points on a pseudo-Riemannian manifold, and \gamma is a geodesic that connects ''p'' and ''q''. Then ''p'' and ''q'' are conjugate points al ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Geodesic
In geometry, a geodesic () is a curve representing in some sense the locally shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. It is a generalization of the notion of a " straight line". The noun '' geodesic'' and the adjective '' geodetic'' come from ''geodesy'', the science of measuring the size and shape of Earth, though many of the underlying principles can be applied to any ellipsoidal geometry. In the original sense, a geodesic was the shortest route between two points on the Earth's surface. For a spherical Earth, it is a segment of a great circle (see also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a geodesic between two vertices/nodes of a graph. In a Riemannian manifold or submanifold, geodesics are characterised by the property of havi ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Spherical Geometry
300px, A sphere with a spherical triangle on it. Spherical geometry or spherics () is the geometry of the two-dimensional surface of a sphere or the -dimensional surface of higher dimensional spheres. Long studied for its practical applications to astronomy, navigation, and geodesy, spherical geometry and the metrical tools of spherical trigonometry are in many respects analogous to Euclidean plane geometry and trigonometry, but also have some important differences. The sphere can be studied either ''extrinsically'' as a surface embedded in 3-dimensional Euclidean space (part of the study of solid geometry), or ''intrinsically'' using methods that only involve the surface itself without reference to any surrounding space. Principles In plane (Euclidean) geometry, the basic concepts are points and (straight) lines. In spherical geometry, the basic concepts are points and great circles. However, two great circles on a plane intersect in two antipodal points, unlike coplan ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Meridian (geography)
In geography and geodesy, a meridian is the locus connecting points of equal longitude, which is the angle (in degrees or other units) east or west of a given prime meridian (currently, the IERS Reference Meridian). In other words, it is a coordinate line for longitudes, a line of longitude. The position of a point along the meridian at a given longitude is given by its latitude, measured in angular degrees north or south of the Equator. On a Mercator projection or on a Gall-Peters projection, each meridian is perpendicular to all circles of latitude. Assuming a spherical Earth, a meridian is a great semicircle on Earth's surface. Adopting instead a spheroidal or ellipsoid model of Earth, the meridian is half of a north-south great ellipse. The length of a meridian is twice the length of an Earth quadrant, equal to on a modern ellipsoid ( WGS 84). Pre-Greenwich The first prime meridian was set by Eratosthenes in 200 BC. This prime meridian was used to provide mea ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Pseudo-Riemannian Manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space that is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible to ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


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. In other words, the Jacobi fields along a geodesic form the tangent space to the geodesic in the space of all geodesics. They are named after Carl Jacobi. Definitions and properties Jacobi fields can be obtained in the following way: Take a smooth one parameter family of geodesics \gamma_\tau with \gamma_0=\gamma, then :J(t)=\left.\frac\_ is a Jacobi field, and describes the behavior of the geodesics in an infinitesimal neighborhood of a given geodesic \gamma. A vector field ''J'' along a geodesic \gamma is said to be a Jacobi field if it satisfies the Jacobi equation: :\fracJ(t)+R(J(t),\dot\gamma(t))\dot\gamma(t)=0, where ''D'' denotes the covariant derivative with respect to the Levi-Civita connection, ''R'' the Riemann curvature tensor, \dot\gamma(t)=d\gamma(t)/dt the tangent vector ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  




Lorentzian Manifold
In mathematical physics, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of Positive-definite bilinear form, positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as Causal structure, timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space that is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of '' ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Congruence (general Relativity)
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation. Types of congruences Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called ''timelike'', ''null'', or ''spacelike'' respectively. A congruence is called a ''geodesic congruence'' if it admits a tangent vector field \vec with vanishing covariant derivative, \nabla_ \vec = 0. Relation with vector fields The integral curves of the vector field are a family of ''non-intersecting'' parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the ''same'' congruenc ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Expansion Parameter
Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansions'' (Lonnie Liston Smith album), 1975 * ''Expansión'' (Mexico), a Mexican news portal linked to CNN * Expansion (sculpture) (2004) Bronze sculpture illuminated from within * ''Expansión'' (Spanish newspaper), a Spanish economic daily newspaper published in Spain * Expansion pack in gaming, extra content for games, often simply "expansion" Science, technology, and mathematics * Expansion (geometry), stretching of geometric objects with flat sides * Expansion (model theory), in mathematical logic, a mutual converse of a reduct * Expansion card, in computing, a printed circuit board that can be inserted into an expansion slot * Expansion chamber, on a two-stroke engine, a tuned exhaust system that enhances power output * Expansion joint ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


Raychaudhuri's Equation
In general relativity, the Raychaudhuri equation, or Landau–Raychaudhuri equation, is a fundamental result describing the motion of nearby bits of matter. The equation is important as a fundamental lemma for the Penrose–Hawking singularity theorems and for the study of exact solutions in general relativity, but has independent interest, since it offers a simple and general validation of our intuitive expectation that gravitation should be a universal attractive force between any two bits of mass–energy in general relativity, as it is in Newton's theory of gravitation. The equation was discovered independently by the Indian physicist Amal Kumar Raychaudhuri and the Soviet physicist Lev Landau. Mathematical statement Given a timelike unit vector field \vec (which can be interpreted as a family or congruence of nonintersecting world lines via the integral curve, not necessarily geodesics), Raychaudhuri's equation in D spacetime dimensions can be written as :\dot = - \frac ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  


picture info

Sphere
A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the center (geometry), ''center'' of the sphere, and the distance is the sphere's ''radius''. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental surface in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. The Earth is spherical Earth, often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]  




Antipodal Point
In mathematics, two points of a sphere (or n-sphere, including a circle) are called antipodal or diametrically opposite if they are the endpoints of a diameter, a straight line segment between two points on a sphere and passing through its center. Given any point on a sphere, its antipodal point is the unique point at greatest distance, whether measured intrinsically (great-circle distance on the surface of the sphere) or extrinsically ( chordal distance through the sphere's interior). Every great circle on a sphere passing through a point also passes through its antipodal point, and there are infinitely many great circles passing through a pair of antipodal points (unlike the situation for any non-antipodal pair of points, which have a unique great circle passing through both). Many results in spherical geometry depend on choosing non-antipodal points, and degenerate if antipodal points are allowed; for example, a spherical triangle degenerates to an underspecified lune if t ...
[...More Info...]      
[...Related Items...]     OR:     [Wikipedia]   [Google]   [Baidu]