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Topic: Finsler manifold

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Banach norm

Riemannian manifold

Riemannian geometry

Geodesic

Intrinsic metric

Metric space

Hausdorff space

Topology

	Manifold - Open Encyclopedia (Site not responding. Last check: 2007-10-08)
		A pseudo-Riemannian manifold is a variant of Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one).
		A symplectic manifold is a manifold equipped with a closed, nondegenerate, alternating 2-form.
		A Kähler manifold is a manifold which simultaneously carries a Riemannian structure, a symplectic structure, and a complex structure which are all compatible in some suitable sense.
	open-encyclopedia.com /Manifold (1809 words)

	Differential geometry and topology
		A vector field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the tangent bundle), such that at each point, the value is an element of the tangent space at that point.
		Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e.
		A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a closed non-degenerate 2-form).
	www.sciencedaily.com /encyclopedia/differential_geometry_and_topology (1043 words)

	Manifold Article, Manifold Information (Site not responding. Last check: 2007-10-08)
		In physics, differentiable manifolds serve as the phase space in classical mechanics and four dimensional pseudo-Riemannian manifolds are used to model spacetime in generalrelativity.
		A pseudo-Riemannian manifold is a variantof Riemannian manifold where the metric tensor is allowed to have an indefinite signature (as opposed to a positive-definite one).
		A Finsler manifold is a generalization of a Riemannianmanifold.
	www.anoca.org /manifolds/space/manifold.html (1294 words)

	Encyclopedia: Riemannian manifold (Site not responding. Last check: 2007-10-08)
		In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point.
		Usually a Riemannian manifold is defined as a smooth manifold with a smooth section of positive-definite quadratic forms on the tangent bundle.
		With this definition of length, every connected Riemannian manifold M becomes a metric space (and even a length metric space) in a natural fashion: the distance d(x, y) between the points x and y of M is defined as
	www.nationmaster.com /encyclopedia/Riemannian-manifold (562 words)

	Finsler geometry -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-08)
		For each point x of M, and for every (A variable quantity that can be resolved into components) vector v in the (additional info and facts about tangent space) tangent space T
		(additional info and facts about Riemannian manifold) Riemannian manifolds (but not (additional info and facts about pseudo Riemannian manifold) pseudo Riemannian manifolds) are special cases of Finsler manifolds.
		The length of γ, a (additional info and facts about differentiable curve) differentiable curve in M is given by
	www.absoluteastronomy.com /encyclopedia/f/fi/finsler_geometry.htm (155 words)

	Finsler geometry in classical mechanics and in Bianchi cosmological models (Site not responding. Last check: 2007-10-08)
		The manifold in which the dynamical systems live is a Finslerian space in which the conformal factor is a positively homogeneous function of first degree in the velocities (the homogeneous Lagrangian of the system).
		Finsler's geometry was developed by Berwald (1926), Eisenhart (1927), Knebelman (1929), Cartan (1934) and Rund (1959) [17].
		In the examples we have considered, the motion of a particle in given potentials, one obtained the motion of a free particle in a curved space (Finsler manifold) whose metric tensor was depending also on the direction (velocity).
	www.sif.it /cimento/tocb/112.02-03/06/06.html (3216 words)

	Z. Shen's papers on Finsler Geometry and Riemannian Geometry
		Then we show that there are topological obstructions for an open manifold to admit a Riemannian metric with quadratic curvature decay and a volume growth which is slower than that of the Euclidean space of the same dimension.
		In this paper, we study Finsler metrics of scalar curvature (i.e., the flag curvature is a scalar function on the slit tangent bundle) and partially determine the flag curvature when certain non-Riemannian quantities are isotropic.
		The flag curvature of a Finsler metric is a natural extension of the sectional curvature in Riemannian geometry, while the S-curvature is a non-Riemannian quantity.
	www.math.iupui.edu /~zshen/Research/preprint1.html (1284 words)

	Finsler manifold - Wikipedia, the free encyclopedia
		In mathematics, a Finsler manifold is a differential manifold M with a Banach norm defined over each tangent space such that the Banach norm as a function of position is smooth, usually it is assumed to satisfy the following regularity condition:
		Riemannian manifolds (but not pseudo-Riemannian manifolds) are special cases of Finsler manifolds.
		The length of γ, a differentiable curve in M, is given by
	en.wikipedia.org /wiki/Finsler_manifold (136 words)

	More on Differential Geometry
		For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rn.
		An alternating k-dimensional linear form is an element of the antisymmetric k'th tensor power of the dual V* of some vector space V. A differential k-form on a manifold is a choice, at each point of the manifold, of such an alternating k-form -- where V is the tangent space at that point.
		Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a 1-form α such that \alpha\wedge (d\alpha)^n does not vanish anywhere.
	www.artilifes.com /differential-geometry.htm (1017 words)

	Differential geometry and topology (Site not responding. Last check: 2007-10-08)
		In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds.
		These all relate to multivariable calculus; but for the geometric applications must be developed in a way that makes good sense without a preferred coordinate system.
		We say a function from the manifold to R is infinitely differentiable if its composition with every homeomorphism results in an infinitely differentiable function from the open unit ball to R.
	www.tocatch.info /en/Differential_topology.htm (1089 words)

	Differential Geometry
		Differential geometry is the study of geometry using calculus.
		In mathematics, contact geometry is the study of completely nonintegrable hyperplane fields on manifolds.
		In mathematics, the differential geometry of curves provides definitions and methods to analyze smooth curves in Riemannian manifolds and Pseudo-Riemannian manifolds (and in particular in Euclidean space) using differential and integral calculus.
	www.differentialgeometry.net (202 words)

	Minimal Entropy Rigidity For Finsler Manifolds Of Negative Flag Curvature (ResearchIndex)
		We define a normalized entropy functional for compact Finsler manifolds of negative flag curvature.
		Using the method of Besson, Courtois, and Gallot, we show that among all such manifolds that are homotopy equivalent to a compact, Riemannian, locally symmetric manifold of negative curvature, the entropy functional is minimized precisely on the locally symmetric manifold.
		2 the dynamics of uniform Finsler manifolds of negative flag c..
	citeseer.ist.psu.edu /416395.html (356 words)

	[No title]
		The definition will depend solely on the fact that the function $L$ is a {\itshape Finsler metric\/} on $\R^n$ (see \cite{Alvarez-Gelfand-Smirnov}): \end{examn} \begin{defi} Let $M$ be a manifold and let $TM\backslash 0$ denote its tangent bundle with the zero section deleted.
		The volume of a Finsler manifold given here is based on the Holmes-Thompson volume of submanifolds of finite-dimensional Banach spaces (see \cite{Holmes-Thompson} and \cite{Thompson}).
		Note that $\Lambda$ inherits a projective Finsler metric from that of $\R^n$ and that $D$ is a hypersurface in $\Lambda$.
	ftp.gwdg.de /EMIS/journals/ERA-AMS/1998-01-013/1998-01-013.tex.html (3411 words)

	Killing equations in classical mechanics (Site not responding. Last check: 2007-10-08)
		The relation between the Killing vectors and tensors admitted by the four-dimensional manifold and the constants of the motion of the geodesics is largely known to people working in the General Relativity theory.
		For a Riemannian manifold of given metric, a vector v on it is a Killing vector if the Lie derivative of the metric along the vector v vanishes [9].
		It is known that it is possible to derive from a Lagrangian also particular dissipative forces depending only on the velocities and, in case, on the time [19, 20] by means of a suitable transformation of the independent variable.
	www.sif.it /cimento/tocb/112.02-03/05/05.html (6640 words)

	iqexpand.com (Site not responding. Last check: 2007-10-08)
		Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a 1-form $\alpha such that \alpha\wedge (d\alpha)^n does not vanish anywhere.$
		Typically, one considers sets which are manifolds (that is, locally resemble Euclidean space) and...
		Subscription Price: US$880.00 (including Sandamp;H)*** Aim and Scope The Journal of Differential Geometry is devoted to the publication of research papers in differential geometry and related subjects such...
	differential_geometry.iqexpand.com (1190 words)

	Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry, Vol. 45, No. 1, pp. 47-59, 2004 (Site not responding. Last check: 2007-10-08)
		Abstract: In this paper we consider a dominating Finsler metric on a complete Riemannian manifold.
		First we prove that the energy integral of the Finsler metric satisfies the Palais-Smale condition, and ask for the number of geodesics with endpoints in two given submanifolds.
		Keywords: Finsler manifold, critical point theory, Palais-Smale condition, Lusternik-Schnirelman theory.
	www.emis.de /journals/BAG/vol.45/no.1/6.html (117 words)

	International Journal of Mathematics and Mathematical Sciences (Site not responding. Last check: 2007-10-08)
		ON THE VERTICAL BUNDLE OF A PSEUDO-FINSLER MANIFOLD
		We define the Liouville distribution on the tangent bundle of a pseudo-Finsler manifold and prove that it is integrable.
		Keywords and phrases: Pseudo-Finsler manifold, Liouville distribution, vertical bundle.
	www.hindawi.com /journals/ijmms/volume-22/S0161171299226373.html (70 words)

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