Relevant bibliographies by topics / Metric geometry of singularities

Academic literature on the topic 'Metric geometry of singularities'

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Published: 1 September 2023

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Journal articles on the topic "Metric geometry of singularities"

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Sabourau, Stéphane, and Zeina Yassine. "A systolic-like extremal genus two surface." Journal of Topology and Analysis 11, no. 03 (September 2019): 721–38. http://dx.doi.org/10.1142/s1793525319500298.

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It is known that the genus two surface admits a piecewise flat metric with conical singularities which is extremal for the systolic inequality among all nonpositively curved metrics. We prove that this piecewise flat metric is also critical for slow metric variations, without curvature restrictions, for another type of systolic inequality involving the lengths of the shortest noncontractible loops in different free homotopy classes.
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Stoica, Ovidiu-Cristinel. "Spacetimes with singularities." Analele Universitatii "Ovidius" Constanta - Seria Matematica 20, no. 2 (June 1, 2012): 213–38. http://dx.doi.org/10.2478/v10309-012-0050-3.

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Abstract We report on some advances made in the problem of singularities in general relativity.First is introduced the singular semi-Riemannian geometry for metrics which can change their signature (in particular be degenerate). The standard operations like covariant contraction, covariant derivative, and constructions like the Riemann curvature are usually prohibited by the fact that the metric is not invertible. The things become even worse at the points where the signature changes. We show that we can still do many of these operations, in a different framework which we propose. This allows the writing of an equivalent form of Einstein's equation, which works for degenerate metric too.Once we make the singularities manageable from mathematical viewpoint, we can extend analytically the black hole solutions and then choose from the maximal extensions globally hyperbolic regions. Then we find space-like foliations for these regions, with the implication that the initial data can be preserved in reasonable situations. We propose qualitative models of non-primordial and/or evaporating black holes.We supplement the material with a brief note reporting on progress made since this talk was given, which shows that we can analytically extend the Schwarzschild and Reissner-Nordström metrics at and beyond the singularities, and the singularities can be made degenerate and handled with the mathematical apparatus we developed.
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Li, Chi. "On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold." Compositio Mathematica 148, no. 6 (October 12, 2012): 1985–2003. http://dx.doi.org/10.1112/s0010437x12000334.

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AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.
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García Ariza, M. Á. "Degenerate Hessian structures on radiant manifolds." International Journal of Geometric Methods in Modern Physics 15, no. 06 (May 8, 2018): 1850087. http://dx.doi.org/10.1142/s0219887818500871.

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We present a rigorous mathematical treatment of Ruppeiner geometry, by considering degenerate Hessian metrics defined on radiant manifolds. A manifold [Formula: see text] is said to be radiant if it is endowed with a symmetric, flat connection and a global vector field [Formula: see text] whose covariant derivative is the identity mapping. A degenerate Hessian metric on [Formula: see text] is a degenerate metric tensor that can locally be written as the covariant Hessian of a function, called potential. A function on [Formula: see text] is said to be extensive if its Lie derivative with respect to [Formula: see text] is the function itself. We show that the Hessian metrics appearing in equilibrium thermodynamics are necessarily degenerate, owing to the fact that their potentials are extensive (up to an additive constant). Manifolds having degenerate Hessian metrics always contain embedded Hessian submanifolds, which generalize the manifolds defined by constant volume in which Ruppeiner geometry is usually studied. By means of examples, we illustrate that linking scalar curvature to microscopic interactions within a thermodynamic system is inaccurate under this approach. In contrast, thermodynamic critical points seem to arise as geometric singularities.
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Birbrair, Lev, and Alexandre Fernandes. "Inner metric geometry of complex algebraic surfaces with isolated singularities." Communications on Pure and Applied Mathematics 61, no. 11 (November 2008): 1483–94. http://dx.doi.org/10.1002/cpa.20244.

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LUO, FENG. "COMBINATORIAL YAMABE FLOW ON SURFACES." Communications in Contemporary Mathematics 06, no. 05 (October 2004): 765–80. http://dx.doi.org/10.1142/s0219199704001501.

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In this paper we develop an approach to conformal geometry of piecewise flat metrics on manifolds. In particular, we formulate the combinatorial Yamabe problem for piecewise flat metrics. In the case of surfaces, we define the combinatorial Yamabe flow on the space of all piecewise flat metrics associated to a triangulated surface. We show that the flow either develops removable singularities or converges exponentially fast to a constant combinatorial curvature metric. If the singularity develops, we show that the singularity is always removable by a surgery procedure on the triangulation. We conjecture that after finitely many such surgery changes on the triangulation, the flow converges to the constant combinatorial curvature metric as time approaches infinity.
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Botvinnik, Boris. "Manifolds with singularities accepting a metric of positive scalar curvature." Geometry & Topology 5, no. 2 (September 26, 2001): 683–718. http://dx.doi.org/10.2140/gt.2001.5.683.

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Larrañaga, Alexis, Natalia Herrera, and Juliana Garcia. "Geometric Description of the Thermodynamics of the Noncommutative Schwarzschild Black Hole." Advances in High Energy Physics 2013 (2013): 1–6. http://dx.doi.org/10.1155/2013/641273.

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The thermodynamics of the noncommutative Schwarzschild black hole is reformulated within the context of the recently developed formalism of geometrothermodynamics (GTD). Using a thermodynamic metric which is invariant with respect to Legendre transformations, we determine the geometry of the space of equilibrium states and show that phase transitions, which correspond to divergencies of the heat capacity, are represented geometrically as singularities of the curvature scalar. This further indicates that the curvature of the thermodynamic metric is a measure of thermodynamic interaction.
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Ashtekar, Abhay, and Javier Olmedo. "Properties of a recent quantum extension of the Kruskal geometry." International Journal of Modern Physics D 29, no. 10 (July 2020): 2050076. http://dx.doi.org/10.1142/s0218271820500765.

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Recently, it was shown that, in an effective description motivated by loop quantum gravity, singularities of the Kruskal spacetime are naturally resolved [A. Ashtekar, J. Olmedo and P. Singh, Phys. Rev. Lett. 121 (2018) 241301; A. Ashtekar, J. Olmedo and P. Singh, Phys. Rev. D 98 (2018) 126003]. In this paper, we explore a few properties of this quantum corrected effective metric. In particular, we (i) calculate the Hawking temperature associated with the horizon of the effective geometry and show that the quantum correction to the temperature is completely negligible for macroscopic black holes, just as one would hope; (ii) discuss the subtleties associated with the asymptotic properties of the spacetime metric, and show that the metric is asymptotically flat in a precise sense; (iii) analyze the asymptotic fall-off of curvature; and, (iv) show that the ADM energy is well defined (and agrees with that determined by the horizon area), even though the curvature falls off less rapidly than in the standard asymptotically flat context.
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Han, Yiwen, and XiaoXiong Zeng. "Legendre Invariance and Geometrothermodynamics Description of the 3D Charged-Dilaton Black Hole." Advances in High Energy Physics 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/865354.

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We first review Weinhold information geometry and Ruppeiner information geometry of 3D charged-dilaton black hole. Then, we use the Legendre invariant to introduce a 2-dimensional thermodynamic metric in the space of equilibrium states, which becomes singular at those points. According to the analysis of the heat capacities, these points are the places where phase transitions occur. This result is valid for the black hole, therefore, provides a geometrothermodynamics description of black hole phase transitions in terms of curvature singularities.
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Dissertations / Theses on the topic "Metric geometry of singularities"

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Oudrane, M'hammed. "Projections régulières, structure de Lipschitz des ensembles définissables et faisceaux de Sobolev." Electronic Thesis or Diss., Université Côte d'Azur, 2023. http://www.theses.fr/2023COAZ4034.

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Dans cette thèse, nous abordons des questions autour de la structure métrique des ensembles définissables dans les structures o-minimales.Dans la première partie, nous étudions les projections régulières au sens de Mostowski, nous prouvons que ces projections n'existent que pour les structures polynomialement bornées, nous utilisons les projections régulières pour refaire la preuve de Parusinski de l'existence des recouvrements réguliers. Dans la deuxième partie de cette thèse, nous étudions les faisceaux de Sobolev (au sens de Lebeau). Pour les fonctions de Sobolev de régularité entière positive, nous construisons ces faisceaux sur le site définissable d'une surface en nous basant sur des observations de base des domaines définissables dans le plan
In this thesis we address questions around the metric structure of definable sets in o-minimal structures. In the first part we study regular projections in the sense of Mostowski, we prove that these projections exists only for polynomially bounded structures, we use regular projections to re perform Parusinski's proof of the existence of regular covers. In the second part of this thesis, we study Sobolev sheaves (in the sense of Lebeau). For Sobolev functions of positive integer regularity, we construct these sheaves on the definable site of a surface based on basic observations of definable domains in the plane
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Lebl, Jiří́. "Singularities and Complexity in CR Geometry." Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2007. http://wwwlib.umi.com/cr/ucsd/fullcit?p3254327.

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Thesis (Ph. D.)--University of California, San Diego, 2007.
Title from first page of PDF file (viewed May 2, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 101-104).
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Ronaldson, Luke James. "The geometry of weak gravitational singularities." Thesis, University of Southampton, 2006. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.485292.

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Coffey, Michael R. "Ricci flow and metric geometry." Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/67924/.

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This thesis considers two separate problems in the field of Ricci flow on surfaces. Firstly, we examine the situation of the Ricci flow on Alexandrov surfaces, which are a class of metric spaces equipped with a notion of curvature. We extend the existence and uniqueness results of Thomas Richard in the closed case to the setting of non-compact Alexandrov surfaces that are uniformly non-collapsed. We complement these results with an extensive survey that collects together, for the first time, the essential topics in the metric geometry of Alexandrov spaces due to a variety of authors. Secondly, we consider a problem in the well-posedness theory of the Ricci flow on surfaces. We show that given an appropriate initial Riemannian surface, we may construct a smooth, complete, immortal Ricci flow that takes on the initial surface in a geometric sense, in contrast to the traditional analytic notions of initial condition. In this way, we challenge the contemporary understanding of well-posedness for geometric equations.
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van, Staden Wernd Jakobus. "Metric aspects of noncommutative geometry." Diss., University of Pretoria, 2019. http://hdl.handle.net/2263/77893.

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We study noncommutative geometry from a metric point of view by constructing examples of spectral triples and explicitly calculating Connes's spectral distance between certain associated pure states. After considering instructive nite-dimensional spectral triples, the noncommutative geometry of the in nite-dimensional Moyal plane is studied. The corresponding spectral triple is based on the Moyal deformation of the algebra of Schwartz functions on the Euclidean plane.
Dissertation (MSc)--University of Pretoria, 2019.
Physics
MSc
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Mangalath, Vishnu. "Singularities of Whitham Deformations." Thesis, The University of Sydney, 2021. https://hdl.handle.net/2123/25990.

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Constant mean curvature planes of finite type in Euclidean 3-space are in correspondence with spectral data, consisting of a hyperelliptic (spectral) curve, two meromorphic differentials, and a line bundle. A class of deformations one can consider are known as Whitham or period preserving deformations. Singularities of Whitham deformations can occur if the differentials have common roots on the spectral curve. In this thesis we are concerned with studying deformations within, and out of, the space of spectral data at which the Whitham equations are singular. We show in a special case that singular Whitham deformations correspond to certain planar graphs on CP1, and study the existence theory of these graphs.
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Palmer, Ian Christian. "Riemannian geometry of compact metric spaces." Diss., Georgia Institute of Technology, 2010. http://hdl.handle.net/1853/34744.

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A construction is given for which the Hausdorff measure and dimension of an arbitrary abstract compact metric space (X, d) can be encoded in a spectral triple. By introducing the concept of resolving sequence of open covers, conditions are given under which the topology, metric, and Hausdorff measure can be recovered from a spectral triple dependent on such a sequence. The construction holds for arbitrary compact metric spaces, generalizing previous results for fractals, as well as the original setting of manifolds, and also holds when Hausdorff and box dimensions differ---in particular, it does not depend on any self-similarity or regularity conditions on the space. The only restriction on the space is that it have positive s₀ dimensional Hausdorff measure, where s₀ is the Hausdorff dimension of the space, assumed to be finite. Also, X does not need to be embedded in another space, such as Rⁿ.
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Jägrell, Linus. "Geometry of the Lunin-Maldacena metric." Thesis, KTH, Teoretisk fysik, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-153502.

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Milicevic, Luka. "Topics in metric geometry, combinatorial geometry, extremal combinatorics and additive combinatorics." Thesis, University of Cambridge, 2018. https://www.repository.cam.ac.uk/handle/1810/273375.

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Persson, Nicklas. "Shortest paths and geodesics in metric spaces." Thesis, Umeå universitet, Institutionen för matematik och matematisk statistik, 2013. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-66732.

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This thesis is divided into three part, the first part concerns metric spaces and specically length spaces where the existence of shortest path between points is the main focus. In the second part, an example of a length space, the Riemannian geometry will be given. Here both a classical approach to Riemannian geometry will be given together with specic results when considered as a metric space. In the third part, the Finsler geometry will be examined both with a classical approach and trying to deal with it as a metric space.
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Books on the topic "Metric geometry of singularities"

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López, Antonio Campillo, and Luis Narváez Macarro, eds. Algebraic Geometry and Singularities. Basel: Birkhäuser Basel, 1996. http://dx.doi.org/10.1007/978-3-0348-9020-5.

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1953-, Campillo Antonio, Narváez Macarro Luis 1957-, and International Conference on Algebraic Geometry (3rd : 1991 : Rábida (Monastery)), eds. Algebraic geometry and singularities. Basel: Birkhäuser Verlag, 1995.

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Snapper, Ernst. Metric affine geometry. New York: Dover Publications, 1989.

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G, Paré E., ed. Descriptive geometry: Metric. 7th ed. New York: Macmillan, 1987.

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Libgober, Anatoly. Trends in Singularities. Basel: Birkhäuser Basel, 2002.

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Blanlœil, Vincent, and Toru Ohmoto, eds. Singularities in Geometry and Topology. Zuerich, Switzerland: European Mathematical Society Publishing House, 2012. http://dx.doi.org/10.4171/118.

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Dai, Xianzhe, and Xiaochun Rong, eds. Metric and Differential Geometry. Basel: Springer Basel, 2012. http://dx.doi.org/10.1007/978-3-0348-0257-4.

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Metric and comparison geometry. Somerville, MA: International Press, 2007.

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Melles, Caroline Grant, and Ruth I. Michler, eds. Singularities in Algebraic and Analytic Geometry. Providence, Rhode Island: American Mathematical Society, 2000. http://dx.doi.org/10.1090/conm/266.

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Neumann, Walter, and Anne Pichon, eds. Introduction to Lipschitz Geometry of Singularities. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-61807-0.

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Book chapters on the topic "Metric geometry of singularities"

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Ruano, Diego. "The Metric Structure of Linear Codes." In Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, 537–61. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-96827-8_24.

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Burago, Yuri, and David Shoenthal. "Metric Geometry." In New Analytic and Geometric Methods in Inverse Problems, 3–50. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. http://dx.doi.org/10.1007/978-3-662-08966-8_1.

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Cambanis, Stamatis, and Donald Richards. "Metric geometry." In I.J. Schoenberg Selected Papers, 189–91. Boston, MA: Birkhäuser Boston, 1988. http://dx.doi.org/10.1007/978-1-4612-3948-2_15.

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de Jong, Theo, and Gerhard Pfister. "Deformations of Singularities." In Local Analytic Geometry, 339–73. Wiesbaden: Vieweg+Teubner Verlag, 2000. http://dx.doi.org/10.1007/978-3-322-90159-0_10.

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de Jong, Theo, and Gerhard Pfister. "Plane Curve Singularities." In Local Analytic Geometry, 171–224. Wiesbaden: Vieweg+Teubner Verlag, 2000. http://dx.doi.org/10.1007/978-3-322-90159-0_5.

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Millman, Richard S., and George D. Parker. "Incidence and Metric Geometry." In Geometry, 17–41. New York, NY: Springer New York, 1991. http://dx.doi.org/10.1007/978-1-4612-4436-3_2.

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Bădescu, Lucian. "Quasi-homogeneous Singularities and Projective Geometry." In Projective Geometry and Formal Geometry, 39–48. Basel: Birkhäuser Basel, 2004. http://dx.doi.org/10.1007/978-3-0348-7936-1_5.

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Brasselet, Jean-Paul. "Singularities and Noncommutative Geometry." In New Developments in Singularity Theory, 135–55. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0834-1_6.

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Pop, Florian. "Alterations and Birational Anabelian Geometry." In Resolution of Singularities, 519–32. Basel: Birkhäuser Basel, 2000. http://dx.doi.org/10.1007/978-3-0348-8399-3_19.

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Hofer, Helmut, Alberto Abbondandolo, Urs Frauenfelder, and Felix Schlenk. "Lagrangian skeleta and plane curve singularities." In Symplectic Geometry, 181–223. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-19111-4_9.

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Conference papers on the topic "Metric geometry of singularities"

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BIRBRAIR, L. "METRIC THEORY OF SINGULARITIES: LIPSCHITZ GEOMETRY OF SINGULAR SPACES." In Proceedings of the Trieste Singularity Summer School and Workshop. WORLD SCIENTIFIC, 2007. http://dx.doi.org/10.1142/9789812706812_0006.

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Ghosal, Ashitava, and Bahram Ravani. "Differential Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators." In ASME 1998 Design Engineering Technical Conferences. American Society of Mechanical Engineers, 1998. http://dx.doi.org/10.1115/detc98/mech-5967.

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Abstract In this paper, we present a differential-geometric analysis of singularities of point trajectories of two and three-degree-of-freedom serial and parallel manipulators. At non-singular configurations, the first order local properties are characterized by the metric coefficients, and, geometrically, by the shape and size of a velocity ellipse and ellipsoid for two and three-degree-of-freedom motions respectively. At singular configurations, the definition of a metric is no longer valid and the velocity ellipsoid degenerates to an ellipse, a line or a point, and the area or the volume of the velocity ellipse or ellipsoid becomes zero. The second and higher order properties, such as curvature, are also not defined at a singularity. In this paper, we use the rate of change of the area or volume to characterize the singularities of the point trajectory. For parallel manipulators, singularities may lead to either loss or gain of one or more degrees-of-freedom. For loss of degree of freedom, the ellipsoid degenerates to an ellipse, a line, or a point as in serial manipulators. For a gain of degree-of-freedom the singularities can be pictured as growth to lines, ellipses, and ellipsoids. The method presented gives a clear geometric picture as to the possible directions and magnitude of motion at a singularity and the local geometry near a singularity. The theoretical results are illustrated with the help of a general spatial 2R manipulator and a three-degree-of-freedom RPSSPR-SPR parallel manipulator.
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VERBITSKY, MISHA. "SINGULARITIES IN HYPERKÄHLER GEOMETRY." In Proceedings of the Second Meeting. WORLD SCIENTIFIC, 2001. http://dx.doi.org/10.1142/9789812810038_0029.

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Chaperon, Marc. "Singularities in contact geometry." In Geometry and Topology of Caustics – Caustics '02. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc62-0-3.

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Goryunov, Victor, and Gabor Lippner. "Simple framed curve singularities." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-6.

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Fukuda, Takuo, and Stanisław Janeczko. "On singularities of Hamiltonian mappings." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-4.

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Berry, M. V. "WAVE GEOMETRY: A PLURALITY OF SINGULARITIES." In Proceedings of the International Conference on Fundamental Aspects of Quantum Theory — to Celebrate 30 Years of the Aharonov-Bohm-Effect. WORLD SCIENTIFIC, 1991. http://dx.doi.org/10.1142/9789814439251_0008.

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Joets, Alain. "Singularities in drawings of singular surfaces." In Geometry and topology of caustics. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2008. http://dx.doi.org/10.4064/bc82-0-10.

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Lu, Zhengdong, Prateek Jain, and Inderjit S. Dhillon. "Geometry-aware metric learning." In the 26th Annual International Conference. New York, New York, USA: ACM Press, 2009. http://dx.doi.org/10.1145/1553374.1553461.

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Ribotta, Roland, Ahmed Belaidi, and Alain Joets. "Singularities, defects and chaos in organized fluids." In Geometry and Topology of Caustics – Caustics '02. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2003. http://dx.doi.org/10.4064/bc62-0-17.

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