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Topic: Finsler metric

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	Title and Abstract
		The Douglas curvature D of Finsler metrics is an important non-Riemannian projective invariant constructed from the Berwald curvature.
		A Finsler metric is called a Douglas metric if its Douglas curvature D. The Douglas metrics are more generalized ones than Berwald metrics and the class of Douglas metrics is much larger than that of Berwald metrics.
		We consider Finsler spaces with a Randers metric $ F = \alpha + \beta $, on the three dimensional real vector space, where $\alpha$ is the Euclidean metric and $\beta$ is a 1-form with norm $b,\,\,0\leq b<1$.
	www.math.iupui.edu /~zshen/Finsler/ISFG/abstract.htm (0 words)

	Springer Online Reference Works (Site not responding. Last check: )
		A metric generalization of Riemannian geometry, where the general definition of the length of a vector is not necessarily given in the form of the square root of a quadratic form as in the Riemannian case.
		is a condition that the Finslerian definition of length is consistent with the particular centro-affine definition; the Finsler metric is needed to compare the lengths of non-collinear vectors.
		The Finsler metric tensor induces a Riemannian metric on the indicatrix, converting it into a Riemannian space.
	eom.springer.de /F/f040390.htm (874 words)

	NationMaster - Encyclopedia: Finsler metric (Site not responding. Last check: )
		The key idea in Finsler geometry is to consider the projectivized tangent bundle PTM (i.e., the bundle of line elements) of the manifold M. The main reason is that all geometric quantities constructed from F are homogeneous of degree zero in y and thus naturally live on PTM, even though F itself does not.
		The Douglas curvature D of Finsler metrics is an important non-Riemannian projective invariant constructed from the Berwald curvature.
		A Finsler metric is called a Douglas metric if its Douglas curvature D. The Douglas metrics are more generalized ones than Berwald metrics and the class of Douglas metrics is much larger than that of Berwald metrics.
	www.nationmaster.com /encyclopedia/Finsler-metric (344 words)

	NationMaster - Encyclopedia: Finsler geometry (Site not responding. Last check: )
		Riemannian manifolds (but not pseudo Riemannian manifolds) are special cases of Finsler manifolds.
		Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves, surfaces were considered as lying in a Euclidean space of higher dimension (for example a surface in an ambient space of three dimensions).
		Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e.
	www.nationmaster.com /encyclopedia/Finsler-geometry (247 words)

	EULER Record Details
		The main idea is to replace the Finsler metric defined on the tangent bundle by a Riemannian metric on a suitable vector subbundle of the tangent bundle of second order, and then to use the standard tools of Riemannian geometry in the study.
		Finsler versions of the Hopf-Rinow, Cartan-Hadamard and Bonnet theorems are proved.\par The second chapter is devoted to the geometry of complex Finsler metrics.
		The Kähler-Finsler metrics and the holomorphic curvature of complex Finsler manifolds are studied.
	www.emis.de /projects/EULER/detail?ide=1994abatfinsmetrglob&matchno=54&matchtotal=190&q=subseries (485 words)

	Publications
		This book is a subindex-free introduction to Finsler geometry that stresses the interactions between metric geometry, the calculus of variations, and convex geometry.
		The notion of isometric submersion is extended to Finsler manifolds and it is used to construct examples of Finsler metrics on complex and quaternionic projective spaces all of whose geodesics are (geometrical) circles.
		Abstract Inspired by Hofer's definition of a metric on the space of compactly supported Hamiltonian maps on a symplectic manifold, this paper exhibits an area-length duality between a class of metric spaces and a class of symplectic manifolds.
	www.math.poly.edu /~alvarez/publications.html (1363 words)

	Finsler Geometry Is Just Riemannian Geometry without the Quadratic Restriction
		The fundamental problem in local Finsler geometry is the equivalence problem: To find a complete system of invariants or to decide when two Finsler metrics differ by a coordinate transformation.
		The key idea in Finsler geometry is to consider the projectivized tangent bundle PTM (i.e., the bundle of line elements) of the manifold M. The main reason is that all geometric quantities constructed from F are homogeneous of degree zero in y and thus naturally live on PTM, even though F itself does not.
		Finsler geometry has been studied from this vantage point by A. Alexandrov [1], H. Busemann [12], and M. Gromov [16].
	www.math.iupui.edu /~zshen/Finsler/history/chern.html (0 words)

	Mathematical Sciences Research Institute - Finsler Geometry
		Finsler geometry uses families of Minkowski norms, instead of families of inner products, to describe geometry.
		These include Finsler spaces of constant curvature, as well as applications of Finsler methods to industrial and medical sectors.
		The purpose of the workshop is to assess the current state of affairs in the field, to provide a forum for technology transfer, and to chart a course for the near future.
	zeta.msri.org /calendar/workshops/WorkshopInfo/195/show_workshop (0 words)

	Hypercomplex.ru
		For the Berwald-Moor type Finsler metric are then considered different types of symmetric polynomials generating the fundamental function and classes of CMC surfaces are evidentiated.
		Further, the metric structures and the metric $N-$linear connections are studied, and the obtained results are specialized to the case when the metric tensor field is of Berwald-Moor type.
		As well, the particular case of $h-$Riemannian $v-$fFBM metric of Riemann-Minkowski type is examined, considering as nonlinear connection both the trivial canonical connection, and the one induced by the Lagrangian of electrodynamics.
	hypercomplex.xpsweb.com /section.php?lang=en (4888 words)

	Hypercomplex.ru
		Bauman Moscow State Technical University invites you to participate in the International Scientific Conference "Finsler extensions of Relativity Theory", to be held in Cairo, Egypt, from 4-10 November 2006.
		Spaces whose metric function is a symmetric polynomial depending on three variables of the third order (the Berwald - Moore metric).
		Spaces whose metric function is a symmetric polynomial depending on four variables of the third order (Chernov spaces).
	hypercomplex.xpsweb.com /section.php?lang=en&genre=8&PHPSESSID=ee6e9a959755f18cf2fc4a449dc4ffa2 (491 words)

	2003 Kemeny Lecture Series
		Abstract: Finsler geometry is a generalization of Riemannian geometry in which one generalizes from having a smoothly varying (positive definite) inner product on the tangent space at each point to having a smoothly varying strictly convex Banach norm on the tangent space at each point.
		A Finsler manifold is said to have constant flag curvature $c$ if its Jacobi operator along any geodesic is conjugate to that along a geodesic in a Riemannian space form of constant sectional curvature $c$.
		In this talk, I will review the basic about Finsler manifolds and their local invariants and discuss what is known about the existence and generality, both local and global, of Finsler metrics of constant flag curvature.
	math.dartmouth.edu /~colloq/s03/bryant.phtml (674 words)

	International Scientific Conference (Site not responding. Last check: )
		The steady growth of interest in the ideas and the program of Finsler Geometry has become increasingly apparent in recent years, and was reflected in the attendance at these two meetings.
		Spaces whose metric function is a symmetric polynomial depending on three variables of the third order (the Berwald√Moore metric).
		Spaces whose metric function is a symmetric polynomial depending on four variables of the fourth order (four-dimensional Berwald√Moore spaces).
	fn.bmstu.ru /phys/konf/ilett_eng.html (531 words)

	On the generalization of Theorems from Riemannian Geometry to Finsler Geometry, , November 27, 2006
		It was performed in [1] the construction of a Riemannian metric and its Levi-Civita's connection in terms of the initial Finsler structure, its associated Chern's connection and the canonical projections associated with it.
		The main procedure is based on the existence of properties and notions which are independent of the âdetailsâ of the Finsler structure, depending only on the Riemannian âskeletonâ and also being true for the initial Finsler structure.
		As consequence, the Cartan-Hadamard theorem and Schur's lemma for Finsler Geometry are proved.
	hermes.aei.mpg.de /arxiv/05/03/704/article.xhtml (2993 words)

	Clearsight Systems Mathlog (Site not responding. Last check: )
		The first connection has a derived affine Finsler metric and is torsion—free.
		Finsler Lagrangians are essential for implementing direct optimization algorithms, maps, control strategies, and adaptive learning.
		Finsler Lagrangians have many nice properties that make them easy to work with and formulate.
	www.clearsightsystems.com /mathlog/default.asp?page=4&sz=5 (444 words)

	Rinton Press - Publisher in Science and Technology
		We show that a lower bound on the minimal size is provided by the length of the minimal geodesic between U and the identity, I, where length is defined by a suitable Finsler metric on the manifold SU(2^n).
		The geodesic curves on these manifolds have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation.
		In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size.
	www.rintonpress.com /journals/qicabstracts/qicabstracts6-3.html (712 words)

	Existence of closed geodesics on positively curved Finsler manifolds, , November 27, 2006
		For non-reversible Finsler metrics of positive flag curvature on spheres and projective spaces we present results about the number and the length of closed geodesics and about their stability properties.
		As a general result we obtain Theorem Â 7Â for metrics on compact and simply-connected manifolds of the rational homotopy type of a compact rank one symmetric space.
		It is mentioned in [BTZ3,p.61]Â that most of the results presented in the Riemannian case generalize to Finsler metrics.
	hermes.aei.mpg.de /arxiv/05/03/153/article.xhtml (2031 words)

	2003 Kemeny Lecture Series
		Abstract: Finsler geometry is a generalization of Riemannian geometry in which one generalizes from having a smoothly varying (positive definite) inner product on the tangent space at each point to having a smoothly varying strictly convex Banach norm on the tangent space at each point.
		A Finsler manifold is said to have constant flag curvature $c$ if its Jacobi operator along any geodesic is conjugate to that along a geodesic in a Riemannian space form of constant sectional curvature $c$.
		In this talk, I will review the basic about Finsler manifolds and their local invariants and discuss what is known about the existence and generality, both local and global, of Finsler metrics of constant flag curvature.
	www.math.dartmouth.edu /~colloq/s03/bryant.phtml (674 words)

	Springer Online Reference Works (Site not responding. Last check: )
		» Encyclopaedia of Mathematics »; F » Finsler metric
		A metric of a space that can be given by a real positive-definite convex function
		A space supplied with a Finsler metric is called a Finsler space, and its geometry Finsler geometry.
	eom.springer.de /f/f040400.htm (80 words)

	Mathematical Sciences Research Institute - Finsler Geometry (Site not responding. Last check: )
		Finsler geometry uses families of Minkowski norms, instead of families of inner products, to describe geometry.
		These include Finsler spaces of constant curvature, as well as applications of Finsler methods to industrial and medical sectors.
		The purpose of the workshop is to assess the current state of affairs in the field, to provide a forum for technology transfer, and to chart a course for the near future.
	www.msri.org /calendar/workshops/WorkshopInfo/195/show_workshop (276 words)

	Publications [#24758] of Robert L Bryant
		A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation.
		A reversible Finsler space is geodesically reversible, but the converse need not be true.
		In this note, building on recent work of LeBrun and Mason, it is shown that a geodesically reversible Finsler metric of constant flag curvature on the 2-sphere is necessarily projectively flat.
	fds.duke.edu /db/aas/math/bryant/publications/24758 (122 words)

	CiteULike: A geometric approach to quantum circuit lower bounds (Site not responding. Last check: )
		This is in contrast with the usual case in circuit design, either classical or quantum, where being given part of an optimal circuit does not obviously assist in the design of the rest of the circuit.
		In this paper we construct several Finsler metrics whose minimal length geodesics provide lower bounds on quantum circuit size, and give a procedure to compute the corresponding geodesic equation.
		The geodesic curves of such a metric have the striking property that once an initial position and velocity are set, the remainder of the geodesic is completely determined by a second order differential equation known as the geodesic equation.
	www.citeulike.org /user/dmitri83/article/101905 (510 words)

	physics - Differential geometry and topology
		Roughly, the contact structure on (2n+1)-dimensional manifold is a choice of a 1-form α such that
		Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, i.e.
		A Finsler metric is much more general structure than a Riemannian metric.
	www.physicsdaily.com /physics/Differential_geometry (973 words)

	Atlas: Critical point theorems of Finsler manifolds by Laszlo Kozma
		In this paper we consider a dominating Finsler metric on a complete Riemannian manifold.
		when a Finsler metric is given on a complete Riemannian manifold.
		We consider only such a Finsler metric which dominates the underlying Riemannian structure of the manifold.
	atlas-conferences.com /c/a/f/h/25.htm (573 words)

	Finsler Geometry and Relativistic Applications
		to extend the ordinary pseudo-Euclidean metric, Hamiltonian and mass-shells;
		will be well stored with various aspects of the relativistic Finsler geometry of space-time.
		FINSLER GEOMETRY, RELATIVITY AND GAUGE THEORIES [D.Reidel, Dordrecht, Holland, 1985]
	www.asanov-finsler.narod.ru (176 words)

	spiro
		, endowed with its Kobayashi metric, is an example of a complex Finsler manifold, which admits a complex geodesic with constant negative curvature through any given point and tangent to any given direction.
		In this talk, we will discuss some recent results concerning the isometric invariants of complex Finsler manifolds and their possible use in the quest for an intrinsic characterization of the domains in
		Faran, V, Hermitian Finsler metrics and the Kobayashi metric, J.
	math.postech.ac.kr /scv-ca/KSCV4/abstract/spiro/spiro.html (156 words)

	CVGMT: The p-Laplace eigenvalue problem in a Finsler metric (Site not responding. Last check: )
		The p-Laplace eigenvalue problem in a Finsler metric
		abstract: We consider the p-Laplacian operator on a domain equipped with a Finsler metric.
		We recall relevant properties of its first eigenfunction for finite p and investigate, in the viscosity solutions setting, the limit problem as p tends to infinity.
	cvgmt.sns.it /news/20040506 (66 words)

	CMB - On Negatively Curved Finsler Manifolds of Scalar Curvature
		In this paper, we prove a global rigidity theorem for negatively curved Finsler metrics on a compact manifold of dimension n \geq 3.
		We show that for such a Finsler manifold, if the flag curvature is a scalar function on the tangent bundle, then the Finsler metric is of Randers type.
		We also study the case when the Finsler metric is locally projectively flat
	journals.cms.math.ca /cgi-bin/vault/view/mo8393?lang=fr (100 words)

	The Field of All Fields
		For example, the restriction to 2nd order, or the assumption of Riemannian metrics rather than, say, Finsler metrics, or the naive assumption of R
		We can compare this to the general theory of relativity, which is compelled by the equivalence principle to represent the metric of spacetime as (so to speak) "the field of all fields".
		defined over a pre-existing metrical space, whose metric we may denote as g.
	www.mathpages.com /rr/s4-06/4-06.htm (2166 words)

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