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\(\mathbf {Nil}\) Geodesic Triangles and Their Interior Angle Sums

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Abstract

In this paper we study the interior angle sums of geodesic triangles in \(\mathbf {Nil}\) geometry and prove that these can be larger, equal or less than \(\pi \). We use for the computations the projective model of \(\mathbf {Nil}\) introduced by Molnár (Beitr. Algebra Geom. 38(2):261–288, 1997).

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References

  • Brodaczewska, K.: Elementargeometrie in \(\mathbf{Nil}\). Dissertation (Dr. rer. nat.) Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden (2014)

  • Chavel, I.: Riemannian Geometry: A Modern Introduction. Cambridge Studies in Advances Mathematics, vol. 98, 2nd edn. Cambridge University Press, Cambridge (2006)

  • Csima, G, Szirmai, J.: Interior angle sum of translation and geodesic triangles in \(\widetilde{\mathbf{S}\mathbf{L}_{2}\mathbf{R}}\) space. arXiv:1610.01500 (2017) (Submitted manuscript)

  • Kobayashi, S., Nomizu, K.: Foundation of Differential Geometry, vol. 1. Inderscience (Wiley) Publishers, New York (1963)

    MATH  Google Scholar 

  • Inoguchi, J.: Minimal translation surfaces in the Heisenberg group \(\mathbf{Nil}_3\). Geom. Dedicata 161(1), 221–231 (2012)

    Article  MathSciNet  Google Scholar 

  • Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  • Molnár, E.: The projective interpretation of the eight 3-dimensional homogeneous geometries. Beitr. Algebra Geom. 38(2), 261–288 (1997)

    MathSciNet  MATH  Google Scholar 

  • Molnár, E.: On projective models of Thurston geometries, some relevant notes on \(\mathbf{Nil}\) orbifolds and manifolds. Sib. Èlektron. Mat. Izv. 7 (2010), 491–498 (2010)

  • Molnár, E., Szirmai, J.: Symmetries in the 8 homogeneous 3-geometries. Symmetry Cult. Sci. 21(1–3), 87–117 (2010)

    MATH  Google Scholar 

  • Molnár, E., Szirmai, J., Vesnin, A.: Projective metric realizations of cone-manifolds with singularities along 2-bridge knots and links. J. Geom. 95, 91–133 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  • Molnár, E., Szirmai, J.: On \(\mathbf{Nil}\) crystallography. Symmetry Cult. Sci. 17(1–2), 55–74 (2006)

    Google Scholar 

  • Pallagi, J., Schultz, B., Szirmai, J.: Equidistant surfaces in \(\mathbf{Nil}\) space. Stud. Univ. Zilina. Math. Ser. 25, 31–40 (2011)

    MathSciNet  MATH  Google Scholar 

  • Schultz, B., Szirmai, J.: On parallelohedra of \(\mathbf{Nil}\)-space. Pollack Periodica 7(Supplement 1), 129–136 (2012)

    Article  Google Scholar 

  • Scott, P.: The geometries of 3-manifolds. Bull. Lond. Math. Soc. 15, 401–487 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  • Szirmai, J.: The densest geodesic ball packing by a type of \(\mathbf{Nil}\) lattices. Beitr. Algebra Geom. 48(2), 383–398 (2007)

    MathSciNet  MATH  Google Scholar 

  • Szirmai, J.: Lattice-like translation ball packings in \(\mathbf{Nil}\) space. Publ. Math. Debrecen 80(3–4), 427–440 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  • Szirmai, J.: A candidate to the densest packing with equal balls in the Thurston geometries. Beitr. Algebra Geom. 55(2), 441–452 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  • Thurston, W. P. (and Levy, S. editor): Three-Dimensional Geometryand Topology, vol. 1 Princeton University Press, Princeton, New Jersey (1997)

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Authors and Affiliations

  1. Department of Geometry, Institute of Mathematics, Budapest University of Technology and Economics, P. O. Box: 91, Budapest, 1521, Hungary

    Jenő Szirmai

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  1. Jenő Szirmai
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Correspondence to Jenő Szirmai.

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Szirmai, J. \(\mathbf {Nil}\) Geodesic Triangles and Their Interior Angle Sums. Bull Braz Math Soc, New Series 49, 761–773 (2018). https://doi.org/10.1007/s00574-018-0077-9

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  • DOI: https://doi.org/10.1007/s00574-018-0077-9

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