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# Decompositions of Teichmüller space by geodesic length mappings by Chen, Min.

Published by Suomalainen Tiedeakatemia, Akateeminen Kirjakauppa [distributor] in Helsinki .

Written in English

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## Subjects:

• Teichmüller spaces.,
• Decomposition (Mathematics)

Edition Notes

Includes bibliographical references (p. 30-31).

## Book details

Classifications The Physical Object Statement Chen Min. Series Annales Academiae Scientiarum Fennicae,, 82 LC Classifications QA331 .C448 1991 Pagination 31 p. : Number of Pages 31 Open Library OL1357762M ISBN 10 9514106636 LC Control Number 92250839

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Decompositions of Teichmüller space by geodesic length mappings. Helsinki: Suomalainen Tiedeakatemia: Akateeminen Kirjakauppa [distributor], (OCoLC) Document Type: Book: All Authors / Contributors: Min Chen. Abstract. The Teichmüller space T(Σ) of a compact C ∞-surface Σ can be parametrized by geodesic length precisely, we can find a set {α1 ,α n} of closed curves α j on Σ such that the isotopy class of a hyperbolic metric d on Σ (i.e.

the point [d] ∊ T(Σ)) is determined by the lengths of geodesic curves homotopic to the curves α j on (Σ, d).Cited by: results about the Teichmuller geodesic ow on moduli space. Along somewhat di erent lines we describe some recent important work of K. Ra that gives a combinatorial description of the Teichmuller metric.

Another important subject is the study of the action of the action of the mapping class group on Teichmuller Size: KB. BEHAVIOR OF GEODESIC-LENGTH FUNCTIONS ON TEICHMULLER SPACE¨ Scott A. Wolpert Abstract Let T be the Teichmul¨ ler space of marked genus g, npunc-tured Riemann surfaces with its bordiﬁcation T the augmented Te-ichmu¨ller space of marked Riemann surfaces with nodes, [Abi77, Ber74].

Provided with the WP metric, T is a complete CAT(0). the geodesic length functions also played an important role in McMullen’s proof of the Kahler hyperbolicity of the moduli space [M].

We want to base our study of geodesic length functions solely upon the hy-perbolic geometry of Riemann surfaces and use the methods of K¨ahler geometry. From this point of view it is desirable to express.

Simple closed geodesics and the study of Teichmuller spaces 3 Property A non-trivial closed curve is freely homotopic to a unique closed geodesic. If the closed curve is simple then so is the freely homotopic closed geodesic.

One way of seeing this is by considering the lifts of a non-trivial curve to the universal cover H. Length of a Curve is Quasi-Convex Along a Teichmüller Geodesic Lenzhen, Anna and Rafi, Kasra, Journal of Differential Geometry, ; Lengths of simple loops on surfaces with hyperbolic metrics Luo, Feng and Stong, Richard, Geometry & Topology, ; The effect of Fenchel-Nielsen coordinates under elementary moves Tan, Dong, Liu, Peijia, and Liu, Xuewen, Kodai Mathematical Journal, Cited by: Geometric mean, splines and de Boor algorithm in geodesic spaces Esfandiar Nava-Yazdani Abstract and Decompositions of Teichmüller space by geodesic length mappings book to as the a ne map of M.

1Some authors use the terminology geodesic length space. Geodesic mappings between Kahler-Weyl spaces and guarantees that „ Fj i is the complex structure of the Kahler-Weyl space KW„ n: As result, we have proved Theorem The KWn admits a nontrivial geodesic mapping onto the Kahler- Weyl space KW„ n, if and only if the following conditions hold a)r_ k„gij = 2(ˆk +Pk)„gij + „gkjˆi + „gikˆj; b)r_ k F„h i= F„hˆi ¡–hF„aˆa.

The maximum deviation of the geodesic from the Decompositions of Teichmüller space by geodesic length mappings book line is near 2, Km and the difference in length is Km.

What this image represents is the actual path taken (geodesic line) if I travel in a straight line, relative to me with no turns, from London to Singapore along the International ellipsoid (this is what I displayed the map in ArcGIS in EPSG).

Purchase Geometry of Riemann Surfaces and Teichmüller Spaces, Volume - 1st Edition. Print Book & E-Book. ISBNBook Edition: 1. Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichm\"uller space, namely the subsets where the lengths of two distinct simple closed geodesics Author: Ursula Hamenstädt.

The space of geodesics may inherit additional structure from the structure of the manifold: for example Hitchin looks at the complex structure of the space of geodesics of ${\mathbb R}^3$ to study monopoles and minimal surfaces in Monopoles and geodesics, Comm.

Math. Phys. Vol Number 4. If a four-dimensional Einstein space with non constant curvature globally admits a geodesic mapping onto a (pseudo-) Riemannian manifold $\bar V_4\in C^1$, then the mapping is affine and, moreover.

CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. We show that both Teichmüller space (with the Teichmüller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces.

For every two geodesic rays in Teichmüller space, we find. Geodesic currents and Teichmuller space" Dragomir Sari&$c∗ USC Department of Mathematics, Kaprielian Hall, RoomLos Angeles, CAUSA Received 27 January ; accepted 4 May Abstract Consider a hyperbolic surface X of innite area. The Liouville map L File Size: KB. As an example, consider the eight-shaped object shown in Fig. 1 and assume that the function f is the elevation (for a generic point (x, y, z) on the object surface f = z).Dark lines shown on the surface of the object highlight some level sets induced by sets correspond to loci of points (x, y, z) on the surface with the same values of the example of Fig. 1, the magnitude of the Cited by: In particular, we give a cell decomposition of the Teichmüller space T(ζ) of (2,1)-surfaces with a boundary geodesic of length 2ζ, for every ζ⩾0; the decomposition is invariant with respect to the mapping class group Γ(2,1) of (2,1)-surfaces. The decomposition has a number of important applications, including one for closed surfaces of Cited by: 5. On the other hand, the mapping class group acts on the 1-skeleton of this complex (see here), and the quotient is a finite graph. So there are finitely many homeomorphism classes of pants decompositions. As far as I can tell, the precise number of homeomorphism classes of pants decompositions is not presently known. See here for a lower bound. in terms of a map from the tangent space T pM to the manifold, this map being deﬁned in terms of geodesics. Deﬁnition Let (M,g)beaRiemannianmani-fold. For every p ∈ M,letD(p)(orsimply,D)bethe open subset of T pM given by D(p)={v ∈ T pM | γ v(1) is deﬁned}, where γ v is the unique maximal geodesic with initial con-ditions γ v. In general relativity, a geodesic generalizes the notion of a "straight line" to curved antly, the world line of a particle free from all external, non-gravitational forces is a particular type of geodesic. In other words, a freely moving or falling particle always moves along a geodesic. In general relativity, gravity can be regarded as not a force but a consequence of a. geodesic path between two simpler trees (Theorem ). This theorem is exploited via dynamic programming and divide and conquer techniques to give two algorithms. Tree Space and Geodesic Distance. This section describes the space of phylogenetic trees, T n, and the geodesic distance. For further details, see [3].Cited by: Example Quicksort. Consider the problem of sorting a sequence A of n elements using the commonly used quicksort algorithm. Quicksort is a divide and conquer algorithm that starts by selecting a pivot element x and then partitions the sequence A into two subsequences A 0 and A 1 such that all the elements in A 0 are smaller than x and all the elements in A 1 are greater than or equal to x. Geometry of Riemann surfaces and Teichmuller spaces / Mika Seppala, Tuomas Sorvali North-Holland ; Distributors for the United States and Canada, Elsevier Science Pub. Co Amsterdam ; New York: New York, N.Y., U.S.A Australian/Harvard Citation. Seppala, Mika. The Geodesic Problem in Quasimetric Spaces thor’s recent study of optimal transport path between probability measures, he ob-serves that there exists a family of very interesting semimetrics on the space of atomic probability measures. These semimetrics satisfy a relaxed triangle inequalityCited by: Teichmu¨ller space. For any isotopy class of closed curves γ, we compute the ﬁrst three derivatives of the length function ℓγ: T(S) →R+ in the shearing coordinates associated to a maximal geodesic lamination λ. We show that if γ intersects each leaf of λ, then the Hessian of ℓγ is positive-deﬁnite. We extend this result to. The area option for geographic coordinates would most likely be geodesic even if the map's display is projected. What it looks like is one thing, what it is, is another. Don't forget, an ellipsoid is only an approximation of Earth's shape, and area and length calculations are best s: to proof, that geodesics minimize the following quantity De nition 2. The value s(t) = Z t t 0 d (t) dt dt= c(t t 0) (3) is called arc length of. It is proportional to the parameter of the geodesic. By setting the value of c= 1, is said to be normalized. To study geodesics, it turns File Size: KB. with pseudo distance 0 one obtains a metric space from it. A geodesic in a metric space is a length minimizing curve parameter-ized proportionally to arclength. A metric space is geodesic if each pair of its points is connected by a geodesic. A subspace of a geodesic space is convexif it is geodesic with respect to the induced metric. A CAT(•). space of curves and show that the generating function for these intersection numbers satisfies the Virasoro equations. We also show that the number of simple closed geodesics of length L on X E Mg,n has the asymptotic behavior sx(L) rv nxL6g-6+n as L We relate the. CiteSeerX - Document Details (Isaac Councill, Lee Giles, Pradeep Teregowda): Abstract. The divergence of two geodesic rays in a metric space that share a basepoint is a function div(t) which is equal to the infimal length of all paths connecting points on these rays that are distance t from the basepoint. We show that the divergence of every two geodesic rays in the Teichmuller space that. On the other hand, I think that Wikipedia might be giving (without saying it) a characterization of rectifiable, not of minimal length. (To define geodesic in a metric space as rectifiable curve of minimal length joining two points). Let me check.$\endgroup\$ – OR. Dec 21 '13 at For a given, a CAT() space is a geodesic metric space in which every geodesic triangle satis es the CAT() inequality.

Such spaces enjoy a rich geometric structure. We will be especially interested in spaces which are CAT(0). Spaces with curvature bounded below; that is, spaces Xfor which inff: Xis a CAT() spaceg>1, have non-bifurcating.

FINDING GEODESICS ON SURFACES TEAM 2: JONGMIN BAEK, ANAND DEOPURKAR, A path γ is a piecewise smooth map deﬁned on I. We will sometime refer to a sequence of points as a path, this refers to the path the minimal geodesic length if we decreased this lower bound on our arc length by choosing u that minimizes LB(u).File Size: 1MB.

a c b l g p p 0 1 Fig The space P(2), showing the geodesic γ and the straight line l between the two points p 0 and p 1. If we consider the matrix Aas a point (a,b,c) ∈ R3, then the above conditions describe the interior of a cone as shown in Fig. Abstract. We show that both Teichm¨uller space (with the Teichm¨uller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of ﬂat spaces and the exponential rate of hyperbolic spaces.

For every two geodesic rays in Teichm¨uller space, we ﬁnd that their divergence. Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichm\"uller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length.

We investigate the point set topology of the union of all such hypersurfaces using elementary methods. New results on the convexity of geodesic-length functions on Teichm¨uller space are presented.

A formula for the Hessian of geodesic-length is presented. New bounds for the gradient and Hessian of geodesic-length are described. A relationship of geodesic-lengthfunctions to Weil-Petersson distance is. The geometry of Teichmüller space via geodesic currents.

[W 4] Wolpert, S.: Geodesic length functions and the Nielsen problem. Differ. Geom, – () Google Scholar; Download references. Author information. by: A complete length space (a metric space where points are connected by geodesics) is not locally compact, i.e.

small balls at a point need not be compact. See for example Complete implies locally compact in length metric space. A good reference for the Arzela-Ascoli thm used here is "Burago et alii: A course in metric geometry".

We show that both Teichmuller space (with the Teichmuller metric) and the mapping class group (with a word metric) have geodesic divergence that is intermediate between the linear rate of flat spaces and the exponential rate of hyperbolic spaces.

For every two geodesic rays in Teichmuller space, we find that their divergence is at most quadratic.A simplified version of this is embodied in Einstein's elevator experiment, illustrated in the figure on the right: for an observer in a small enclosed room, it is impossible to decide, by mapping the trajectory of bodies such as a dropped ball, whether the room is at rest in a gravitational field, or in free space aboard a rocket that is accelerating at a rate equal to that of the gravitational field.Place cells and grid cells.

Pyramidal neurons in the rat hippocampus have long been known to have firing fields in localized areas of space.While much research has studied hippocampal neurons with small place fields, – (e.g., roughly the size of a rat) a range of place field scales have been reported.Recently, electrophysiological recordings from a long linear track suggest that Cited by:

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