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Journal articles on the topic 'Riemannian and barycentric geometry'

Author: Grafiati

Published: 13 February 2024

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1

Pihajoki, Pauli, Matias Mannerkoski, and Peter H. Johansson. "Barycentric interpolation on Riemannian and semi-Riemannian spaces." Monthly Notices of the Royal Astronomical Society 489, no. 3 (September 2, 2019): 4161–69. http://dx.doi.org/10.1093/mnras/stz2447.

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ABSTRACT Interpolation of data represented in curvilinear coordinates and possibly having some non-trivial, typically Riemannian or semi-Riemannian geometry is a ubiquitous task in all of physics. In this work, we present a covariant generalization of the barycentric coordinates and the barycentric interpolation method for Riemannian and semi-Riemannian spaces of arbitrary dimension. We show that our new method preserves the linear accuracy property of barycentric interpolation in a coordinate-invariant sense. In addition, we show how the method can be used to interpolate constrained quantities so that the given constraint is automatically respected. We showcase the method with two astrophysics related examples situated in the curved Kerr space–time. The first problem is interpolating a locally constant vector field, in which case curvature effects are expected to be maximally important. The second example is a general relativistic magnetohydrodynamics simulation of a turbulent accretion flow around a black hole, wherein high intrinsic variability is expected to be at least as important as curvature effects.

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2

Miranda Jr., Gastão F., Gilson Giraldi, Carlos E. Thomaz, and Daniel Millàn. "Composition of Local Normal Coordinates and Polyhedral Geometry in Riemannian Manifold Learning." International Journal of Natural Computing Research 5, no. 2 (April 2015): 37–68. http://dx.doi.org/10.4018/ijncr.2015040103.

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The Local Riemannian Manifold Learning (LRML) recovers the manifold topology and geometry behind database samples through normal coordinate neighborhoods computed by the exponential map. Besides, LRML uses barycentric coordinates to go from the parameter space to the Riemannian manifold in order to perform the manifold synthesis. Despite of the advantages of LRML, the obtained parameterization cannot be used as a representational space without ambiguities. Besides, the synthesis process needs a simplicial decomposition of the lower dimensional domain to be efficiently performed, which is not considered in the LRML proposal. In this paper, the authors address these drawbacks of LRML by using a composition procedure to combine the normal coordinate neighborhoods for building a suitable representational space. Moreover, they incorporate a polyhedral geometry framework to the LRML method to give an efficient background for the synthesis process and data analysis. In the computational experiments, the authors verify the efficiency of the LRML combined with the composition and discrete geometry frameworks for dimensionality reduction, synthesis and data exploration.

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3

Sabatini, Luca. "Volume Comparison in the presence of a Gromov-Hausdorff ε−approximation II." Annals of West University of Timisoara - Mathematics and Computer Science 56, no. 1 (July 1, 2018): 99–135. http://dx.doi.org/10.2478/awutm-2018-0008.

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Abstract Let (M, g) be any compact, connected, Riemannian manifold of dimension n. We use a transport of measures and the barycentre to construct a map from (M, g) onto a Hyperbolic manifold (ℍn/Λ, g0) (Λ is a torsionless subgroup of Isom(ℍn,g0)), in such a way that its jacobian is sharply bounded from above. We make no assumptions on the topology of (M, g) and on its curvature and geometry, but we only assume the existence of a measurable Gromov-Hausdorff ε-approximation between (ℍn/Λ, g0) and (M, g). When the Hausdorff approximation is continuous with non vanishing degree, this leads to a sharp volume comparison, if $\varepsilon < {1 \over {64\,{n^2}}}\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)$ , then $$\matrix{{Vol\left( {{M^n},g} \right) \ge }\cr {{{\left( {1 + 160n\left( {n + 1} \right)\sqrt {{\varepsilon \over {\min \left( {in{j_{\left( {{{\Bbb H}^n}/\Lambda ,{g_0}} \right)}},1} \right)}}} } \right)}^{{n \over 2}}}\left| {\deg \,h} \right| \cdot Vol\left( {{X^n},{g_0}} \right).} \cr }$$

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4

Wu, H., and Wilhelm Klingenberg. "Riemannian Geometry." American Mathematical Monthly 92, no. 7 (August 1985): 519. http://dx.doi.org/10.2307/2322529.

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5

Lord, Nick, M. P. do Carmo, S. Gallot, D. Hulin, J. Lafontaine, I. Chavel, and D. Martin. "Riemannian Geometry." Mathematical Gazette 79, no. 486 (November 1995): 623. http://dx.doi.org/10.2307/3618122.

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6

Mrugała, R. "Riemannian geometry." Reports on Mathematical Physics 27, no. 2 (April 1989): 283–85. http://dx.doi.org/10.1016/0034-4877(89)90011-6.

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7

M.Osman, Mohamed. "Differentiable Riemannian Geometry." International Journal of Mathematics Trends and Technology 29, no. 1 (January 25, 2016): 45–55. http://dx.doi.org/10.14445/22315373/ijmtt-v29p508.

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8

Dimakis, Aristophanes, and Folkert Müller-Hoissen. "Discrete Riemannian geometry." Journal of Mathematical Physics 40, no. 3 (March 1999): 1518–48. http://dx.doi.org/10.1063/1.532819.

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9

Beggs, Edwin J., and Shahn Majid. "Poisson–Riemannian geometry." Journal of Geometry and Physics 114 (April 2017): 450–91. http://dx.doi.org/10.1016/j.geomphys.2016.12.012.

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10

Strichartz, Robert S. "Sub-Riemannian geometry." Journal of Differential Geometry 24, no. 2 (1986): 221–63. http://dx.doi.org/10.4310/jdg/1214440436.

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11

Rylov, Yuri A. "Geometry without topology as a new conception of geometry." International Journal of Mathematics and Mathematical Sciences 30, no. 12 (2002): 733–60. http://dx.doi.org/10.1155/s0161171202012243.

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A geometric conception is a method of a geometry construction. The Riemannian geometric conception and a new T-geometric one are considered. T-geometry is built only on the basis of information included in the metric (distance between two points). Such geometric concepts as dimension, manifold, metric tensor, curve are fundamental in the Riemannian conception of geometry, and they are derivative in the T-geometric one. T-geometry is the simplest geometric conception (essentially, only finite point sets are investigated) and simultaneously, it is the most general one. It is insensitive to the space continuity and has a new property: the nondegeneracy. Fitting the T-geometry metric with the metric tensor of Riemannian geometry, we can compare geometries, constructed on the basis of different conceptions. The comparison shows that along with similarity (the same system of geodesics, the same metric) there is a difference. There is an absolute parallelism in T-geometry, but it is absent in the Riemannian geometry. In T-geometry, any space region is isometrically embeddable in the space, whereas in Riemannian geometry only convex region is isometrically embeddable. T-geometric conception appears to be more consistent logically, than the Riemannian one.

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12

Greene, Robert E. "Book Review: Riemannian geometry." Bulletin of the American Mathematical Society 21, no. 1 (July 1, 1989): 157–63. http://dx.doi.org/10.1090/s0273-0979-1989-15802-3.

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13

Kunzinger, Michael, and Roland Steinbauer. "Generalized pseudo-Riemannian geometry." Transactions of the American Mathematical Society 354, no. 10 (June 3, 2002): 4179–99. http://dx.doi.org/10.1090/s0002-9947-02-03058-1.

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14

Larsen, Jens Chr. "Singular semi-riemannian geometry." Journal of Geometry and Physics 9, no. 1 (February 1992): 3–23. http://dx.doi.org/10.1016/0393-0440(92)90023-t.

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15

Jesus, W. D. R., and A. F. Santos. "On causality violation in Lyra Geometry." International Journal of Geometric Methods in Modern Physics 15, no. 08 (June 22, 2018): 1850143. http://dx.doi.org/10.1142/s0219887818501438.

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In this paper, the causality issues are discussed in a non-Riemannian geometry, called Lyra geometry. It is a non-Riemannian geometry originated from Weyl geometry. In order to compare this geometry with the Riemannian geometry, the Einstein field equations are considered. It is verified that the Gödel and Gödel-type metric are consistent with this non-Riemannian geometry. A non-trivial solution for Gödel universe in the absence of matter sources is determined without analogue in general relativity. Different sources are considered and then different conditions for causal and non-causal solutions are discussed.

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16

Ciaglia, F. M., G. Marmo, and J. M. Pérez-Pardo. "Generalized potential functions in differential geometry and information geometry." International Journal of Geometric Methods in Modern Physics 16, supp01 (January 29, 2019): 1940002. http://dx.doi.org/10.1142/s0219887819400024.

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Potential functions can be used for generating potentials of relevant geometric structures for a Riemannian manifold such as the Riemannian metric and affine connections. We study whether this procedure can also be applied to tensors of rank four and find a negative answer. We study this from the perspective of solving the inverse problem and also from an intrinsic point of view.

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17

Chinor, Makur Makuac, Mohamed Y. A. Bakhet, and Abdelmajid Ali Dafallah. "On Some Role of Riemannian Geometry." Asian Research Journal of Mathematics 19, no. 5 (March 25, 2023): 51–60. http://dx.doi.org/10.9734/arjom/2023/v19i5659.

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This paper examines the mathematical significance of Riemannian geometry, including surfaces, Riemannian curvature, Gauss curvature, manifolds, geodesics, and the relationships between them. It also explores the applications of Riemannian geometry in various concepts.

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18

TAVAKOL, REZA. "GEOMETRY OF SPACETIME AND FINSLER GEOMETRY." International Journal of Modern Physics A 24, no. 08n09 (April 10, 2009): 1678–85. http://dx.doi.org/10.1142/s0217751x09045224.

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A central assumption in general relativity is that the underlying geometry of spacetime is pseudo-Riemannian. Given the recent attempts at generalizations of general relativity, motivated both by theoretical and observational considerations, an important question is whether the spacetime geometry can also be made more general and yet still remain compatible with observations? Here I briefly summarize some earlier results which demonstrate that there are special classes of Finsler geometry, which is a natural metrical generalization of the Riemannian geometry, that are strictly compatible with the observations regarding the motion of idealised test particles and light rays. I also briefly summarize some recent attempts at employing Finsler geometries motivated by more recent developments such as those in String theory, whereby Lorentz invariance is partially broken.

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19

BEJANCU, AUREL. "A LINEAR CONNECTION FOR BOTH SUB-RIEMANNIAN GEOMETRY AND NONHOLONOMIC MECHANICS (I)." International Journal of Geometric Methods in Modern Physics 08, no. 04 (June 2011): 725–52. http://dx.doi.org/10.1142/s0219887811005361.

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We study the geometry of a sub-Riemannian manifold (M, HM, VM, g), where HM and VM are the horizontal and vertical distribution respectively, and g is a Riemannian extension of the Riemannian metric on HM. First, without the assumption that HM and VM are orthogonal, we construct a sub- Riemannian connection ▽ on HM and prove some Bianchi identities for ▽. Then, we introduce the horizontal sectional curvature, prove a Schur theorem for sub-Riemannian geometry and find a class of sub-Riemannian manifolds of constant horizontal curvature. Finally, we define the horizontal Ricci tensor and scalar curvature, and some sub-Riemannian differential operators (gradient, divergence, Laplacian), extending some results from geometry to the sub-Riemannian setting.

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20

Lefevre, Jeanne, Florent Bouchard, Salem Said, Nicolas Le Bihan, and Jonathan H. Manton. "On Riemannian and non-Riemannian Optimisation, and Optimisation Geometry." IFAC-PapersOnLine 54, no. 9 (2021): 578–83. http://dx.doi.org/10.1016/j.ifacol.2021.06.119.

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21

Sari, Ramazan, and Mehmet Akyol. "Hemi-slant ξ⊥-Riemannian submersions in contact geometry." Filomat 34, no. 11 (2020): 3747–58. http://dx.doi.org/10.2298/fil2011747s.

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M. A. Akyol and R. Sar? [On semi-slant ??-Riemannian submersions, Mediterr. J. Math. 14(6) (2017) 234.] defined semi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. As a generalization of the above notion and natural generalization of anti-invariant ??-Riemannian submersions, semi-invariant ??-Riemannian submersions and slant submersions, we study hemi-slant ??-Riemannian submersions from Sasakian manifolds onto Riemannian manifolds. We obtain the geometry of foliations, give some examples and find necessary and sufficient condition for the base manifold to be a locally product manifold. Moreover, we obtain some curvature relations from Sasakian space forms between the total space, the base space and the fibres.

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22

Nazimuddin, AKM, and Md Showkat Ali. "Riemannian Geometry and Modern Developments." GANIT: Journal of Bangladesh Mathematical Society 39 (November 19, 2019): 71–85. http://dx.doi.org/10.3329/ganit.v39i0.44159.

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In this paper, we compute the Christoffel Symbols of the first kind, Christoffel Symbols of the second kind, Geodesics, Riemann Christoffel tensor, Ricci tensor and Scalar curvature from a metric which plays a fundamental role in the Riemannian geometry and modern differential geometry, where we consider MATLAB as a software tool for this implementation method. Also we have shown that, locally, any Riemannian 3-dimensional metric can be deformed along a directioninto another metricthat is conformal to a metric of constant curvature GANIT J. Bangladesh Math. Soc.Vol. 39 (2019) 71-85

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23

Boscain, Ugo, Remco Duits, Francesco Rossi, and Yuri Sachkov. "Curve cuspless reconstructionviasub-Riemannian geometry." ESAIM: Control, Optimisation and Calculus of Variations 20, no. 3 (May 27, 2014): 748–70. http://dx.doi.org/10.1051/cocv/2013082.

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24

Hitchin, Nigel. "Integral systems in Riemannian geometry." Surveys in Differential Geometry 4, no. 1 (1998): 21–81. http://dx.doi.org/10.4310/sdg.1998.v4.n1.a1.

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25

Wolf, Michael. "TEICHMÜLLER THEORY IN RIEMANNIAN GEOMETRY." Bulletin of the London Mathematical Society 26, no. 3 (May 1994): 315–16. http://dx.doi.org/10.1112/blms/26.3.315.

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26

Bejancu, Aurel. "Curvature in sub-Riemannian geometry." Journal of Mathematical Physics 53, no. 2 (February 2012): 023513. http://dx.doi.org/10.1063/1.3684957.

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27

Chen, Bang-Yen. "RIEMANNIAN GEOMETRY OF LAGRANGIAN SUBMANIFOLDS." Taiwanese Journal of Mathematics 5, no. 4 (December 2001): 681–723. http://dx.doi.org/10.11650/twjm/1500574989.

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28

Beggs, Edwin, and Shahn Majid. "Nonassociative Riemannian geometry by twisting." Journal of Physics: Conference Series 254 (November 1, 2010): 012002. http://dx.doi.org/10.1088/1742-6596/254/1/012002.

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29

Petersen, Peter. "Aspects of global Riemannian geometry." Bulletin of the American Mathematical Society 36, no. 03 (May 24, 1999): 297–345. http://dx.doi.org/10.1090/s0273-0979-99-00787-9.

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30

Chaichian, M., A. Tureanu, R. B. Zhang, and Xiao Zhang. "Riemannian geometry of noncommutative surfaces." Journal of Mathematical Physics 49, no. 7 (July 2008): 073511. http://dx.doi.org/10.1063/1.2953461.

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31

Puetzfeld, Dirk, and Yuri N. Obukhov. "Probing non-Riemannian spacetime geometry." Physics Letters A 372, no. 45 (November 2008): 6711–16. http://dx.doi.org/10.1016/j.physleta.2008.09.041.

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32

Brandt, Howard E. "Riemannian geometry of quantum computation." Nonlinear Analysis: Theory, Methods & Applications 71, no. 12 (December 2009): e474-e486. http://dx.doi.org/10.1016/j.na.2008.11.013.

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33

Nikonorov, Yu G., E. D. Rodionov, and V. V. Slavskii. "Geometry of homogeneous Riemannian manifolds." Journal of Mathematical Sciences 146, no. 6 (November 2007): 6313–90. http://dx.doi.org/10.1007/s10958-007-0472-z.

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34

Lord, Steven, Adam Rennie, and Joseph C. Várilly. "Riemannian manifolds in noncommutative geometry." Journal of Geometry and Physics 62, no. 7 (July 2012): 1611–38. http://dx.doi.org/10.1016/j.geomphys.2012.03.004.

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35

Majid, Shahn. "Noncommutative Riemannian geometry on graphs." Journal of Geometry and Physics 69 (July 2013): 74–93. http://dx.doi.org/10.1016/j.geomphys.2013.02.004.

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36

Boucetta, Mohamed. "Riemannian geometry of Lie algebroids." Journal of the Egyptian Mathematical Society 19, no. 1-2 (April 2011): 57–70. http://dx.doi.org/10.1016/j.joems.2011.09.009.

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37

Ho, Pei-Ming. "Riemannian Geometry on Quantum Spaces." International Journal of Modern Physics A 12, no. 05 (February 20, 1997): 923–43. http://dx.doi.org/10.1142/s0217751x97000694.

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An algebraic formulation of Riemannian geometry on quantum spaces is presented, where Riemannian metric, distance, Laplacian, connection, and curvature have their counterparts. This description is also extended to complex manifolds. Examples include the quantum sphere, the complex quantum projective space and the two-sheeted space.

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38

DI SCALA, ANTONIO J., ANDREA LOI, and FABIO ZUDDAS. "RIEMANNIAN GEOMETRY OF HARTOGS DOMAINS." International Journal of Mathematics 20, no. 02 (February 2009): 139–48. http://dx.doi.org/10.1142/s0129167x09005236.

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Let DF = {(z0, z) ∈ ℂn | |z0|2 < b, ||z||2 < F(|z0|2)} be a strongly pseudoconvex Hartogs domain endowed with the Kähler metric gF associated to the Kähler form [Formula: see text]. This paper contains several results on the Riemannian geometry of these domains. These are summarized in Theorems 1.1–1.3. In the first one we prove that if DF admits a non-special geodesic (see definition below) through the origin whose trace is a straight line then DF is holomorphically isometric to an open subset of the complex hyperbolic space. In the second theorem we prove that all the geodesics through the origin of DF do not self-intersect, we find necessary and sufficient conditions on F for DF to be geodesically complete and we prove that DF is locally irreducible as a Riemannian manifold. Finally, in Theorem 1.3, we compare the Bergman metric gB and the metric gF in a bounded Hartogs domain and we prove that if gB is a multiple of gF, namely gB = λ gF, for some λ ∈ ℝ+, then DF is holomorphically isometric to an open subset of the complex hyperbolic space.

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39

Borisenko, A. A., and A. L. Yampol'skii. "Riemannian geometry of fibre bundles." Russian Mathematical Surveys 46, no. 6 (December 31, 1991): 55–106. http://dx.doi.org/10.1070/rm1991v046n06abeh002859.

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40

Agrachev, A. A. "Topics in sub-Riemannian geometry." Russian Mathematical Surveys 71, no. 6 (December 31, 2016): 989–1019. http://dx.doi.org/10.1070/rm9744.

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41

Wu, H. "Riemannian Geometry. By Wilhelm Klingenberg." American Mathematical Monthly 92, no. 7 (August 1985): 519–22. http://dx.doi.org/10.1080/00029890.1985.11971671.

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42

Strichartz, Robert S. "Corrections to: ``Sub-Riemannian geometry''." Journal of Differential Geometry 30, no. 2 (1989): 595–96. http://dx.doi.org/10.4310/jdg/1214443604.

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43

Chanillo, Sagun, and B. Muckenhoupt. "Nodal geometry on Riemannian manifolds." Journal of Differential Geometry 34, no. 1 (1991): 85–91. http://dx.doi.org/10.4310/jdg/1214446991.

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44

Klartag, Bo’az. "Needle Decompositions in Riemannian Geometry." Memoirs of the American Mathematical Society 249, no. 1180 (September 2017): 0. http://dx.doi.org/10.1090/memo/1180.

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45

Branson, T. "Kato constants in Riemannian geometry." Mathematical Research Letters 7, no. 3 (2000): 245–61. http://dx.doi.org/10.4310/mrl.2000.v7.n3.a1.

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46

Brendle, Simon. "Evolution equations in Riemannian geometry." Japanese Journal of Mathematics 6, no. 1 (September 2011): 45–61. http://dx.doi.org/10.1007/s11537-011-1115-1.

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47

Fang, Fuquan, Xiaochun Rong, Wilderich Tuschmann, and Yihu Yang. "Topics in Metric Riemannian Geometry." Frontiers of Mathematics in China 11, no. 5 (August 12, 2016): 1097–98. http://dx.doi.org/10.1007/s11464-016-0581-4.

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48

Hernández, Gerardo, and Ernesto A. Lacomba. "Contact Riemannian geometry and thermodynamics." Differential Geometry and its Applications 8, no. 3 (June 1998): 205–16. http://dx.doi.org/10.1016/s0926-2245(98)00006-0.

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49

Nhikawa, Seiki, Philippe Tondeur, and Lieven Vanhecke. "Spectral geometry for Riemannian foliations." Annals of Global Analysis and Geometry 10, no. 3 (1992): 291–304. http://dx.doi.org/10.1007/bf00136871.

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50

López, Jesús A. Álvarez, Yuri A. Kordyukov, and Eric Leichtnam. "Riemannian foliations of bounded geometry." Mathematische Nachrichten 287, no. 14-15 (April 9, 2014): 1589–608. http://dx.doi.org/10.1002/mana.201300211.

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