Relevant bibliographies by topics / Invariant Riemannian metrics

Academic literature on the topic 'Invariant Riemannian metrics'

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Published: 10 March 2023

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Journal articles on the topic "Invariant Riemannian metrics"

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Wang, Hui, and Shaoqiang Deng. "Left Invariant Einstein–Randers Metrics on Compact Lie Groups." Canadian Mathematical Bulletin 55, no. 4 (December 1, 2012): 870–81. http://dx.doi.org/10.4153/cmb-2011-145-6.

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AbstractIn this paper we study left invariant Einstein–Randers metrics on compact Lie groups. First, we give a method to construct left invariant non-Riemannian Einstein–Randers metrics on a compact Lie group, using the Zermelo navigation data. Then we prove that this gives a complete classification of left invariant Einstein–Randers metrics on compact simple Lie groups with the underlying Riemannian metric naturally reductive. Further, we completely determine the identity component of the group of isometries for this type of metrics on simple groups. Finally, we study some geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature of such metrics.
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Parhizkar, M., and D. Latifi. "On the flag curvature of invariant (α,β)-metrics." International Journal of Geometric Methods in Modern Physics 13, no. 04 (March 31, 2016): 1650039. http://dx.doi.org/10.1142/s0219887816500390.

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In this paper, we consider invariant [Formula: see text]-metrics which are induced by invariant Riemannian metrics [Formula: see text] and invariant vector fields [Formula: see text] on homogeneous spaces. We study the flag curvatures of invariant [Formula: see text]-metrics. We first give an explicit formula for the flag curvature of invariant [Formula: see text]-metrics arising from invariant Riemannian metrics on homogeneous spaces and Lie groups. We then give some explicit formula for the flag curvature of invariant Matsumoto metrics, invariant Kropina metrics and invariant Randers metrics.
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Balashchenko, V. V., P. N. Klepikov, E. D. Rodionov, and O. P. Khromova. "On the Cerbo Conjecture on Lie Groups with the Left-Invariant Lorentzian Metric." Izvestiya of Altai State University, no. 1(123) (March 18, 2022): 79–82. http://dx.doi.org/10.14258/izvasu(2022)1-12.

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Manifolds with constraints on tensor fields include Einstein manifolds, Einstein-like manifolds, conformally flat manifolds, and a number of other important classes of manifolds. The work of many mathematicians is devoted to the study of such manifolds, which is reflected in the monographs of A. Besse, M. Berger, M.-D. Cao, M. Wang. Ricci solitons are one of the natural generalizations of Einstein's metrics. If a Riemannian manifold is a Lie group, one speaks of invariant Ricci solitons. Invariant Ricci solitons were studied in most detail in the case of unimodular Lie groups with left-invariant Riemannian metrics and the case of low dimension. Thus, L. Cerbo proved that all invariant Ricci solitons are trivial on unimodular Lie groups with left-invariant Riemannian metric and Levi-Civita connection.A similar result up to dimension four was obtained by P.N. Klepikov and D.N. Oskorbin for the non-unimodular case. We study invariant Ricci solitons on three-dimensional unimodular Lie groups with the Lorentzian metric.The study results show that unimodular Lie groups with left-invariant Lorentzian metric admit invariant Ricci solitons other than trivial ones. In this paper, a complete classification of invariant Ricci solitons on three-dimensional unimodular Lie groups with leftinvariant Lorentzian metric is obtained.
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Hashinaga, Takahiro, and Hiroshi Tamaru. "Three-dimensional solvsolitons and the minimality of the corresponding submanifolds." International Journal of Mathematics 28, no. 06 (May 2, 2017): 1750048. http://dx.doi.org/10.1142/s0129167x17500483.

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In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional simply-connected solvable Lie groups are completely characterized by the minimality of the corresponding submanifolds.
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Asgari, Farhad, and Hamid Reza Salimi Moghaddam. "Left invariant Randers metrics of Berwald type on tangent Lie groups." International Journal of Geometric Methods in Modern Physics 15, no. 01 (December 19, 2017): 1850015. http://dx.doi.org/10.1142/s0219887818500159.

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Let [Formula: see text] be a Lie group equipped with a left invariant Randers metric of Berward type [Formula: see text], with underlying left invariant Riemannian metric [Formula: see text]. Suppose that [Formula: see text] and [Formula: see text] are lifted Randers and Riemannian metrics arising from [Formula: see text] and [Formula: see text] on the tangent Lie group [Formula: see text] by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold [Formula: see text] and the sectional curvature of the Riemannian manifold [Formula: see text] when [Formula: see text] is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors.
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chen, Chao, Zhiqi chen, and Yuwang Hu. "Einstein metrics and Einstein–Randers metrics on a class of homogeneous manifolds." International Journal of Geometric Methods in Modern Physics 15, no. 04 (March 13, 2018): 1850052. http://dx.doi.org/10.1142/s0219887818500524.

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In this paper, we give [Formula: see text]-invariant Einstein metrics on a class of homogeneous manifolds [Formula: see text], and then prove that every homogeneous manifold [Formula: see text] admits at least three families of [Formula: see text]-invariant non-Riemannian Einstein–Randers metrics.
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Arvanitoyeorgos, Andreas, V. V. Dzhepko, and Yu G. Nikonorov. "Invariant Einstein Metrics on Some Homogeneous Spaces of Classical Lie Groups." Canadian Journal of Mathematics 61, no. 6 (December 1, 2009): 1201–13. http://dx.doi.org/10.4153/cjm-2009-056-2.

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Abstract A Riemannian manifold (M, ρ) is called Einstein if the metric ρ satisfies the condition Ric(ρ) = c · ρ for some constant c. This paper is devoted to the investigation of G-invariant Einstein metrics, with additional symmetries, on some homogeneous spaces G/H of classical groups. As a consequence, we obtain new invariant Einstein metrics on some Stiefel manifolds SO(n)/SO(l). Furthermore, we show that for any positive integer p there exists a Stiefelmanifold SO(n)/SO(l) that admits at least p SO(n)-invariant Einstein metrics.
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Vylegzhanin, D. V., P. N. Klepikov, E. D. Rodionov, and O. P. Khromova. "On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton." Izvestiya of Altai State University, no. 4(120) (September 10, 2021): 86–90. http://dx.doi.org/10.14258/izvasu(2021)4-13.

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Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians. Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case. This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.
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Deng, Shaoqiang, and Zixin Hou. "Invariant Randers metrics on homogeneous Riemannian manifolds." Journal of Physics A: Mathematical and General 39, no. 18 (April 19, 2006): 5249–50. http://dx.doi.org/10.1088/0305-4470/39/18/c01.

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Deng, Shaoqiang, and Zixin Hou. "Invariant Randers metrics on homogeneous Riemannian manifolds." Journal of Physics A: Mathematical and General 37, no. 15 (March 29, 2004): 4353–60. http://dx.doi.org/10.1088/0305-4470/37/15/004.

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Dissertations / Theses on the topic "Invariant Riemannian metrics"

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Vasconcelos, Rosa Tayane de. "O tensor de Ricci e campos de killing de espaços simétricos." reponame:Repositório Institucional da UFC, 2017. http://www.repositorio.ufc.br/handle/riufc/25968.

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VASCONCELOS, Rosa Tayane de. O tensor de Ricci e campos de killing de espaços simétricos. 2017. 81 f. Dissertação (Mestrado em Matemática)- Centro de Ciências, Universidade Federal do Ceará, Fortaleza, 2017.
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This work brings a smooth and self-contained introduction to the study of the most basic aspects of symmetric spaces, having as its nal goal the characterization of the Killing vector fields and of the Ricci tensor of such riemannian manifolds. Several of the results presented in the initial chapter are not easily found, in the Diferential Geometry literature, in a way as accessible and self-contained as here. This being said, we believe that this work embodies some didactic relevance, for it others students interested in symmetric spaces a relatively smooth first contact. We shall generally look at symmetric spaces as homogeneous manifolds G=H, where G is a Lie group and H is a closed Lie subgroup of G, such that the natural mapping : G ! G=H is a riemannian submersion. Ultimately, this map allows us to describe the relationships between the curvature, the Ricci tensor and the geodesics of G and G=H. For our purposes, the crucial remark is that, under appropriate circumstances, one guarantees the existence, in G=H, of a metric for which left translations are isometries. Hence, a one-parameter family of such isometries gives rise to a Killing vector field, which turn into a Jacobi vector eld when restricted to a geodesic. We present explicit expressions for such Jacobi vector elds, showing that they only depend on the eigenvalues of the linear operator TX : g ! g given by TX = (adX)2, for certain vector elds X 2 g.
Este trabalho traz uma introdução suave e autocontida ao estudo dos aspectos mais básicos de espaços simétricos, tendo como objetivo final a caracterização dos campos de Killing e do tensor de Ricci de tais variedades riemannianas. Vários dos resultados obtidos nos capítulos iniciais não são encontrados, na literatura de Geometria Diferencial, de maneira tão acessível e autocontida como apresentados aqui. Com isso, acreditamos que o trabalho reveste-se de alguma relevância didática, por oferecer aos alunos interessados no estudo de espaços simétricos um primeiro contato relativamente suave. Em linhas gerais, veremos espaços simétricos como variedades homogêneas G=H, onde G e um grupo de Lie e H um subgrupo de Lie fechado de G, tais que a aplicação natural: G ! G=H seja uma submersão riemanniana. Através dela, descrevemos relações entre a curvatura, o tensor de Ricci e as geodésicas de G e G=H. Para nossos propósitos, a observação crucial e que, sob certas hipóteses, garantimos a existência, em G=H, de uma métrica cujas translações a esquerda são isometrias. Portanto, uma família a um parâmetro de tais isometrias d a origem a um campo de Killing que, por sua vez, restrito a geodésicas torna-se um campo de Jacobi. Apresentamos expressões para esses campos de Jacobi, mostrando que os mesmos só dependem dos autovalores do operador linear TX : g ! g dado por TX = (adX)2, para certos campos X 2 g.
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Karki, Manoj Babu. "Invariant Riemannain metrics on four-dimensional Lie group." University of Toledo / OhioLINK, 2015. http://rave.ohiolink.edu/etdc/view?acc_num=toledo1438906778.

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Alekseevsky, Dmitri, Andreas Kriegl, Mark Losik, Peter W. Michor, and Peter Michor@esi ac at. "The Riemannian Geometry of Orbit Spaces. The Metric, Geodesics, and." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi997.ps.

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Becker, Christian. "On the Riemannian geometry of Seiberg-Witten moduli spaces." Phd thesis, [S.l. : s.n.], 2005. http://deposit.ddb.de/cgi-bin/dokserv?idn=975744771.

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Pediconi, Francesco. "Geometric aspects of locally homogeneous Riemannian spaces." Doctoral thesis, 2020. http://hdl.handle.net/2158/1197175.

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The subject of this thesis is the study of some geometric problems arising in the context of locally and globally homogeneous Riemannian spaces. In particular, we are mainly interested in investigate the interplay between curvature conditions and the compactness of some classes of locally homogeneous spaces, with respect to appropriate topologies.
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Books on the topic "Invariant Riemannian metrics"

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An Introduction to Extremal Kahler Metrics. Providence, Rhode Island: Springer, 2014.

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Book chapters on the topic "Invariant Riemannian metrics"

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Tamaru, Hiroshi. "The Space of Left-Invariant Riemannian Metrics." In Springer Proceedings in Mathematics & Statistics, 315–26. Tokyo: Springer Japan, 2016. http://dx.doi.org/10.1007/978-4-431-56021-0_17.

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Alekseevskii, D. V., and B. A. Putko. "On the completeness of left-invariant pseudo-Riemannian metrics on lie groups." In Lecture Notes in Mathematics, 171–85. Berlin, Heidelberg: Springer Berlin Heidelberg, 1990. http://dx.doi.org/10.1007/bfb0085954.

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Leutwiler, Heinz. "A riemannian metric invariant under Möbius transformations in ℝn." In Lecture Notes in Mathematics, 223–35. Berlin, Heidelberg: Springer Berlin Heidelberg, 1988. http://dx.doi.org/10.1007/bfb0081257.

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Bahadır, Oguzhan. "Curvature Tensors of Screen Semi-invariant Half-Lightlike Submanifolds of a Semi-Riemannian Product Manifold with Quarter-Symmetric Non-metric Connection." In Mathematical Methods and Modelling in Applied Sciences, 136–46. Cham: Springer International Publishing, 2020. http://dx.doi.org/10.1007/978-3-030-43002-3_13.

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Fefferman, Charles, and C. Robin Graham. "Jet Isomorphism." In The Ambient Metric (AM-178). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691153131.003.0008.

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A fundamental result in Riemannian geometry is the jet isomorphism theorem which asserts that at the origin in geodesic normal coordinates, the full Taylor expansion of the metric may be recovered from the iterated covariant derivatives of curvature. As a consequence, one deduces that any local invariant of Riemannian metrics has a universal expression in terms of the curvature tensor and its covariant derivatives. Geodesic normal coordinates are determined up to the orthogonal group, so problems involving local invariants are reduced to purely algebraic questions concerning invariants of the orthogonal group on tensors. This chapter proves an analogous jet isomorphism theorem for conformal geometry. By making conformal changes, the Taylor expansion of a metric in geodesic normal coordinates can be further simplified, resulting in a “conformal normal form” for metrics about a point.
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Fefferman, Charles, and C. Robin Graham. "Scalar Invariants." In The Ambient Metric (AM-178). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691153131.003.0009.

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This chapter shows how to derive a characterization of scalar invariants of conformal structures by reduction to the relevant results of [BEGr]. In [FG], the authors conjectured that when n is odd, all scalar conformal invariants arise as Weyl invariants constructed from the ambient metric. The second main goal of this book is to prove this together with an analogous result when n is even. These results are contained in Theorems 9.2, 9.3, and 9.4. The parabolic invariant theory needed to prove these results was developed in [BEGr], including the observation of the existence of exceptional invariants. But substantial work is required to reduce the theorems in the chapter to the results of [BEGr]. To understand this, it briefly reviews how Weyl's characterization of scalar Riemannian invariants is proved.
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Fefferman, Charles, and C. Robin Graham. "Introduction." In The Ambient Metric (AM-178). Princeton University Press, 2011. http://dx.doi.org/10.23943/princeton/9780691153131.003.0001.

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This introductory chapter begins with a brief definition of conformal geometry. Conformal geometry is the study of spaces in which one knows how to measure infinitesimal angles but not lengths. A conformal structure on a manifold is an equivalence class of Riemannian metrics, in which two metrics are identified if one is a positive smooth multiple of the other. In [FG], the authors outlined a construction of a nondegenerate Lorentz metric in n+2 dimensions associated to an n-dimensional conformal manifold, which they called the ambient metric. This association enables one to construct conformal invariants in n dimensions from pseudo-Riemannian invariants in n+2 dimensions, and in particular shows that conformal invariants are plentiful. The formal theory outlined in [FG] did not provide details. This book provides these details. An overview of the subsequent chapters is also presented.
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Nolte, David D. "Relativistic Dynamics." In Introduction to Modern Dynamics, 385–425. Oxford University Press, 2019. http://dx.doi.org/10.1093/oso/9780198844624.003.0012.

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The invariance of the speed of light with respect to any inertial observational frame leads to a surprisingly large number of unusual results that defy common intuition. Chief among these are time dilation, length contraction, and loss of simultaneity. The Lorentz transformation intermixes space and time, but an overarching structure is provided by the metric tensor of Minkowski space-time. The pseudo-Riemannian metric supports 4-vectors whose norms are invariants, independent of any observational frame. These invariants constitute the proper objects of reality to study in the special theory of relativity. Relativistic dynamics defines the equivalence of mass and energy, which has many applications in nuclear energy and particle physics. Forces have transformation properties between relatively moving frames that set the stage for a more general theory of relativity that describes physical phenomena in noninertial frames.
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Tu, Loring W. "Integration on a Compact Connected Lie Group." In Introductory Lectures on Equivariant Cohomology, 103–14. Princeton University Press, 2020. http://dx.doi.org/10.23943/princeton/9780691191751.003.0013.

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This chapter explores integration on a compact connected Lie group. One of the great advantages of working with a compact Lie group is the possibility of extending the notion of averaging from a finite group to the compact Lie group. If the compact Lie group is connected, then there exists a unique bi-invariant top-degree form with total integral 1, which simplifies the presentation of averaging. The averaging operator is useful for constructing invariant objects. For example, suppose a compact connected Lie group G acts smoothly on the left on a manifold M. Given any C∞ differential k-form ω‎ on M, by averaging all the left translates of ω‎ over G, one can produce a C∞ invariant k-form on M. As another example, on a G-manifold one can average all translates of a Riemannian metric to produce an invariant Riemann metric.
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"Contact Metric Manifolds and Submanifolds." In Pseudo-Riemannian Geometry, δ-Invariants and Applications, 241–50. WORLD SCIENTIFIC, 2011. http://dx.doi.org/10.1142/9789814329644_0012.

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Conference papers on the topic "Invariant Riemannian metrics"

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Zhengwu Zhang, Eric Klassen, Anuj Srivastava, Pavan Turaga, and Rama Chellappa. "Blurring-invariant Riemannian metrics for comparing signals and images." In 2011 IEEE International Conference on Computer Vision (ICCV). IEEE, 2011. http://dx.doi.org/10.1109/iccv.2011.6126442.

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Zhang, Yi, and Kwun-Lon Ting. "Point-Line Distance Under Riemannian Metrics." In ASME 2005 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. ASMEDC, 2005. http://dx.doi.org/10.1115/detc2005-84637.

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A point-line is the combination of a directed line and an endpoint on the line. A pair of point-line positions corresponds to a point-line displacement, which is known to be associated with a set of rigid body displacements whose screw axes are distributed on a cylindroid. Different associated rigid body displacements generally correspond to different distances under Riemannian metrics on the manifold of SE(3). A unique measure of distance between a pair of point-line positions is desirable in engineering applications. In this paper, the distance between two point-line positions is investigated based on the left-invariant Riemannian metrics on the manifold of SE(3). The displacements are elaborated from the perspective of the soma space. The set of rigid body displacements associated to the point-line displacement is mapped to a one-dimensional great circle on the unit sphere in the space of four dual dimensions, on which the point with the minimum distance to the identity is indicated. It is shown that the minimum distance is achieved when an associated rigid body displacement has no rotational component about the point-line axis. The minimum distance, which has the inherited property of independence of inertial reference frames, is referred to as the point-line distance. A numerical example shows the application of point-line distance to a point-line path generation problem in mechanism synthesis.
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Berestovskii, Valerii Nikolaevich. "Geodesics and curvatures of left-invariant sub-Riemannian metrics on Lie groups." In International Conference "Optimal Control and Differential Games" dedicated to the 110th anniversary of L. S. Pontryagin. Moscow: Steklov Mathematical Institute, 2018. http://dx.doi.org/10.4213/proc22961.

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Ilea, Ioana, Lionel Bombrun Bombrun, Salem Said, and Yannick Berthoumieu. "Covariance Matrices Encoding Based on the Log-Euclidean and Affine Invariant Riemannian Metrics." In 2018 IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW). IEEE, 2018. http://dx.doi.org/10.1109/cvprw.2018.00080.

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Park, Frank C. "A Geometric Framework for Optimal Surface Design." In ASME 1992 Design Technical Conferences. American Society of Mechanical Engineers, 1992. http://dx.doi.org/10.1115/detc1992-0171.

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Abstract We present a Riemannian geometric framework for variational approaches to geometric design. Optimal surface design is regarded as a special case of the more general problem of finding a minimum distortion mapping between Riemannian manifolds. This geometric approach emphasizes the coordinate-invariant aspects of the problem, and engineering constraints are naturally embedded by selecting a suitable metric in the physical space. In this context we also present an engineering application of the theory of harmonic maps.
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Paraskevopoulos, Elias, and Sotirios Natsiavas. "On a Consistent Application of Newton’s Law to Constrained Mechanical Systems." In ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2013. http://dx.doi.org/10.1115/detc2013-12346.

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An investigation is carried out for deriving conditions on the correct application of Newton’s law of motion to mechanical systems subjected to constraints. It utilizes some fundamental concepts of differential geometry and treats both holonomic and anholonomic constraints. The focus is on establishment of conditions, so that the form of Newton’s law remains invariant when imposing an additional set of motion constraints on a system. Based on this requirement, two conditions are derived, specifying the metric and the form of the connection on the new manifold. The latter is weaker than a similar condition employed frequently in the literature, but holding on Riemannian manifolds only. This is shown to have several practical implications. First, it provides a valuable freedom for selecting the connection on the manifold describing large rigid body rotation, so that the group properties of this manifold are preserved. Moreover, it is used to state clearly the conditions for expressing Newton’s law on the tangent space (and not on the dual space) of a manifold. Finally, the Euler-Lagrange operator is examined and issues related to equations of motion for anholonomic and vakonomic systems are investigated.
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