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Year 2024, Volume: 17 Issue: 1, 232 - 244, 23.04.2024

Stefan Haesen Miroslava Petrović-torgašev Leopold Verstraelen

https://doi.org/10.36890/iejg.1466330

Abstract

References

  • [1] Albujer, A. and Haesen, S.: A geometrical interpretation of the null sectional curvature. J. Geom. Phys. 60, 471-476(2010).
  • [2] Belkhelfa, M., Deszcz, R. and Verstraelen, L.: Symmetry properties of 3-dimensional d’Atri spaces. Kyungpook Math. J. 46, 367-376 (2006).
  • [3] Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003).
  • [4] Berger, M.: La géométrie métrique de variétés riemanniennes (...), in “Élie Cartan et les mathématiques d’ aujourd’hui”, Astérisque, Paris, 9- 66(1985).
  • [5] Boeckx, E., Kowalski, O. and Vanhecke, L.: Riemannian manifolds of conullity two. World Scientific, Singapore (1996).
  • [6] Boju, B. and Popescu, M.: Espaces à courbure quasi-constante. J. Diff. Geom. 13, 373-383(1978).
  • [7] Chen, B.-Y.: Geometry of Submanifolds, Marcel Dekker, New York (1973).
  • [8] Chen, B.-Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications. World Scientific, Singapore (2011).
  • [9] Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific, Singapore (2017).
  • [10] Chen, B.-Y. and Yano, K.: Hypersurfaces of a conformally flat space. Tensor N.S. 26, 318-322 (1972).
  • [11] Chen, B.-Y. and Yano, K.:Special conformally flat spaces and canal hypersurfaces. Tôhoku Math. J. 25, 177-184(1973).
  • [12] Chen, B.-Y., Dillen, F., Verstraelen, L. and Vrancken, L.: Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces. Proc. AMS 128, 589-598 (1999).
  • [13] Defever, F. and Deszcz, R.: A note on geodesic mappings of pseudosymmetric Riemannian manifolds. Colloq. Math. 62, 313-319(1991).
  • [14] Decu, S., Janahara, B., Petrovic-Torgasev, M. and Verstraelen, L.: On the Chen character of δ(2) ideal submanifolds. Kragujevac Math. J. 32, 37-46 (2009).
  • [15] Decu, S., Pantc, A., Petrovic-Torgasev, M. and Verstraelen, L.: Ricci and Casorati principal directions of δ(2) Chen ideal submanifolds. Kragujevac Math. J. 37, 25-31 (2013).
  • [16] Deprez, J., Deszcz, R. and Verstraelen,L.: Examples of pseudo-symmetric conformally flat warped products. Chinese J. Math. 17, 51-65 (1989).
  • [17] Deszcz, R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. A 44, 1-34(1992).
  • [18] Deszcz, R., Verstraelen, L. and Yaprak,¸S.: Pseudo-symmetric hypersurfaces in 4-dimensional spaces of constant curvature. Bull. Inst. Math. Acad. Sinica, 22, 167-179 (1994).
  • [19] Deszcz, R., Verstraelen, L. and Yaprak, Ş.: Warped products realizing a certain condition of pseudosymmetry type imposed on the Weyl curvature tensor. Chinese J. Math. 22, 139-157(1994).
  • [20] Deszcz, R., Haesen, S. and Verstraelen, L.: Chapter 6: On Natural Symmetries. in Topics in Differential Geometry, (eds. A. & I. Mihai and R. Miron), Ed. Acad. Române, Bucharest, 249-308(2008).
  • [21] Dillen, F., Fastenakels, J., Haesen, S., Van der Veken, J. and Verstraelen, L.: Submanifold theory and Parallel transport. Kragujevac J. Math. 37, 33-43(2013).
  • [22] Dombrowski, P.: 150 years after Gauss’ ‘’Disquisitiones generalis circa superficias curvas. Astérisque, Paris (1979).
  • [23] Eisenhart, L.P.: Riemannian Geometry. Princeton University Press, Princeton (edition 1997).
  • [24] Freudenthal, H. and Steiner, H.-G.: Chapter 13: Group Theory and Geometry, in Fundamentals of Mathematics, Vol. II: Geometry. (eds. H. Behnke e.a.), MIT Press, Cambridge, 516-533 (1974).
  • [25] Ganchev, G. and Mihova, V.: Riemannian manifolds of quasi-constant sectional curvature. J. reine angew. Math. 522, 119-141(2000).
  • [26] Ganchev, G. and Mihova, V.: A classification of Riemannian manifolds of quasi-constant sectional curvatures. Ann. Sofia Univ. Fac. Math. and Inf. 102, 115-138 (2015).
  • [27] Gunawan, H., Neswan, O. and Setya-Budhi, W.: A formule for angles between subspaces of inner product spaces. Beitr. Algebra Geom. 46, 311-320 (2009).
  • [28] Haesen, S. and Verstraelen, L.: Properties of a scalar curvature invariant depending on two planes. Manuscripta Math. 122, 59-72(2007).
  • [29] Haesen, S. and Verstraelen, L.: Natural Intrinsic Geometrical Symmetries. SIGMA 5, pp. 15 (2009). Special Issue ‘’Élie Cartan and Differential Geometry.
  • [30] Haesen, S. and Verstraelen, L.: Chapter 8: Curvatures and Symmetries of Parallel Transport, and Chapter 9: Extrinsic Symmetries of Parallel transport. in Differential Geometry and Topology, Discrete and Computational Geometry, (eds. J.-M. Morvan and M. Boucetta), NATO Science Series - IOS Press, 197-255(2005).
  • [31] Jahanara, B., Haesen, S., ¸Sentürk, Z. and Verstraelen, L.: On the parallel transport of the Ricci curvatures. J. Geom. Phys. 57, 1771-1777 (2007).
  • [32] Klein, F.: Elementary mathematics from an advanced standpoint - Geometry, Dover, New York (1939).
  • [33] Kühnel, W.: Differentialgeometrie. Kurven - Flächen - Mannigfaltigheiten. Vieweg, Wiesbaden (2008); English translation: Differential Geometry. Curves - Surfaces - Manifolds. AMS Student Mathematical Library 16 (2006).
  • [34] Kowalski, O. and Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three - elliptic case. Rend. Mat. Appl. 17(7), 477-512( 1997).
  • [35] Kowalski, O. and Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three - non-elliptic case. Bull. Tokyo Gakugei Univ. 50(4), 1-28(1998).
  • [36] Levy, H.: Tensors determined by a hypersurface in Riemannian space. Trans. AMS 28, 671-694(1926).
  • [37] Lumiste, Ü.: Semi-parallel submanifolds in real space forms. Springer, Berlin (2009).
  • [38] Matveev, V.S.: Geometric explanation of the Beltrami theorem. Int. J. Methods Mod. Phys. 3, 623-629(2006).
  • [39] Mikeš, J., Kiosak, V. and Vanžurová, A.: Geodesic mappings of manifolds with affine connection. Palacký University, Olomouc (2008).
  • [40] Mikeš, J., Vanžurová, A. and Hinterleitner, I.: Geodesic mappings and some generalizations. Palacký University, Olomouc (2009).
  • [41] Risteki, I.B. and Trencevski, K.: Principal values and principal subspaces of two subspaces of vector spaces with inner product. Beitr. Algebra Geom. 42, 289-300(2001).
  • [42] Schouten, J.A.: Die direkte Analysis zur neueren Relativitätstheorie. Verhan. Konink. Akad. Wet. Amsterdam 12, nr. 6, 1-95(1918).
  • [43] Sekigawa, K. and Takagi, H.: On the conformally flat spaces satisfying a curvature condition on the Ricci tensor. Tôhoku Math. J. 23, 1-11 (1971).
  • [44] Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. I. The local version. J. Diff. Geom. 17 , 531-582 (1982).
  • [45] Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. II. The global version. Geom. Dedicata 19, 65-108(1985).
  • [46] Szabó, Z. I.: A short topological proof of the symmetry of 2-point homogeneous spaces. Invent. math. 106, 61-64 (1991).
  • [47] Thurston, W.: The Geometry and Topology of 3-Manifolds., Lecture Notes, Princeton University Press, Princeton (1979).
  • [48] Verstraelen, L.: Natural extrinsic geometrical symmetries - an introduction. AMS Contempory Mathematics, Vol. 674 5-16(2016). (from the ‘’AMS Special Session on Recent Advances in the Geometry of Submanifolds Dedicated to the Memory of Franki Dillen (1963-2013)”, (eds. B. Suceava e.a.).
  • [49] Verstraelen, L.: Comments on the pseudo-symmetry in the sense of Ryszard Deszcz. Geometry and Topology of Submanifolds VI, (eds. F. Dillen e.a.), World Scientific, Singapore, 199-209(1994).
  • [50] Verstraelen, L.: Submanifold theory - A contemplation of submanifolds, AMS Contemporary Mathematics, Vol. 756, 21-56(2020); (from the ‘’AMS Special Session on the Geometry of Submanifolds in Honor of Bang-Yen Chen”, (eds. J. Van der Veken e.a.)).
  • [51] Wall, C. T. C.: Geometries and Geometrical Structures in the real dimension 4 and the complex dimension 2. Springer Lecture Notes 1167, eds. Dold and Eckmann; Proc. Univ. of Maryland Special Year of Low Dimensional Topology , College Park, (1983-1984).
  • [52] Weyl, H.: Symmetry. Princeton University Press, Princeton (1952).
  • [53] Yano, K., Houh, C.-S. and Chen, B.-Y.: Intrinsic characterization of certain conformally flat spaces. Kodai Math. Sem. Rep. 25, 357-361(1973).

On Thurston's Geometrical Space Form Problem: On Quasi Space Forms

Year 2024, Volume: 17 Issue: 1, 232 - 244, 23.04.2024

Stefan Haesen Miroslava Petrović-torgašev Leopold Verstraelen

https://doi.org/10.36890/iejg.1466330

Abstract

A proposal is made for what may well be the most elementary Riemannian spaces which are
homogeneous but not isotropic. In other words: a proposal is made for what may well be the
nicest symmetric spaces beyond the real space forms, that is, beyond the Riemannian spaces which
are homogeneous and isotropic. The above qualification of ‘’nicest symmetric spaces” finds a
justification in that, together with the real space forms, these spaces are most natural with respect to
the importance in human vision of our ability to readily recognise conformal things and in that these
spaces are most natural with respect to what inWeyl’s view is symmetry in Riemannian geometry.
Following his suggestion to remove the real space forms’ isotropy condition, the quasi space forms
thus introduced do offer a metrical, local geometrical solution to the geometrical space form problem
as posed by Thurston in his 1979 Princeton Lecture Notes on ‘’The Geometry and Topology of 3-
manifolds”. Roughly speaking, quasi space forms are the Riemannian manifolds of dimension
greater than or equal to 3, which are not real space forms but which admit two orthogonally
complementary distributions such that at all points all the 2-planes that in the tangent spaces there
are situated in a same position relative to these distributions do have the same sectional curvatures.

Keywords

Quasi space forms Riemannian geometry Deszcz symmetric spaces

References

  • [1] Albujer, A. and Haesen, S.: A geometrical interpretation of the null sectional curvature. J. Geom. Phys. 60, 471-476(2010).
  • [2] Belkhelfa, M., Deszcz, R. and Verstraelen, L.: Symmetry properties of 3-dimensional d’Atri spaces. Kyungpook Math. J. 46, 367-376 (2006).
  • [3] Berger, M.: A Panoramic View of Riemannian Geometry. Springer, Berlin (2003).
  • [4] Berger, M.: La géométrie métrique de variétés riemanniennes (...), in “Élie Cartan et les mathématiques d’ aujourd’hui”, Astérisque, Paris, 9- 66(1985).
  • [5] Boeckx, E., Kowalski, O. and Vanhecke, L.: Riemannian manifolds of conullity two. World Scientific, Singapore (1996).
  • [6] Boju, B. and Popescu, M.: Espaces à courbure quasi-constante. J. Diff. Geom. 13, 373-383(1978).
  • [7] Chen, B.-Y.: Geometry of Submanifolds, Marcel Dekker, New York (1973).
  • [8] Chen, B.-Y.: Pseudo-Riemannian Geometry, δ-Invariants and Applications. World Scientific, Singapore (2011).
  • [9] Chen, B.-Y.: Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific, Singapore (2017).
  • [10] Chen, B.-Y. and Yano, K.: Hypersurfaces of a conformally flat space. Tensor N.S. 26, 318-322 (1972).
  • [11] Chen, B.-Y. and Yano, K.:Special conformally flat spaces and canal hypersurfaces. Tôhoku Math. J. 25, 177-184(1973).
  • [12] Chen, B.-Y., Dillen, F., Verstraelen, L. and Vrancken, L.: Characterizations of Riemannian space forms, Einstein spaces and conformally flat spaces. Proc. AMS 128, 589-598 (1999).
  • [13] Defever, F. and Deszcz, R.: A note on geodesic mappings of pseudosymmetric Riemannian manifolds. Colloq. Math. 62, 313-319(1991).
  • [14] Decu, S., Janahara, B., Petrovic-Torgasev, M. and Verstraelen, L.: On the Chen character of δ(2) ideal submanifolds. Kragujevac Math. J. 32, 37-46 (2009).
  • [15] Decu, S., Pantc, A., Petrovic-Torgasev, M. and Verstraelen, L.: Ricci and Casorati principal directions of δ(2) Chen ideal submanifolds. Kragujevac Math. J. 37, 25-31 (2013).
  • [16] Deprez, J., Deszcz, R. and Verstraelen,L.: Examples of pseudo-symmetric conformally flat warped products. Chinese J. Math. 17, 51-65 (1989).
  • [17] Deszcz, R.: On pseudosymmetric spaces. Bull. Soc. Math. Belg. A 44, 1-34(1992).
  • [18] Deszcz, R., Verstraelen, L. and Yaprak,¸S.: Pseudo-symmetric hypersurfaces in 4-dimensional spaces of constant curvature. Bull. Inst. Math. Acad. Sinica, 22, 167-179 (1994).
  • [19] Deszcz, R., Verstraelen, L. and Yaprak, Ş.: Warped products realizing a certain condition of pseudosymmetry type imposed on the Weyl curvature tensor. Chinese J. Math. 22, 139-157(1994).
  • [20] Deszcz, R., Haesen, S. and Verstraelen, L.: Chapter 6: On Natural Symmetries. in Topics in Differential Geometry, (eds. A. & I. Mihai and R. Miron), Ed. Acad. Române, Bucharest, 249-308(2008).
  • [21] Dillen, F., Fastenakels, J., Haesen, S., Van der Veken, J. and Verstraelen, L.: Submanifold theory and Parallel transport. Kragujevac J. Math. 37, 33-43(2013).
  • [22] Dombrowski, P.: 150 years after Gauss’ ‘’Disquisitiones generalis circa superficias curvas. Astérisque, Paris (1979).
  • [23] Eisenhart, L.P.: Riemannian Geometry. Princeton University Press, Princeton (edition 1997).
  • [24] Freudenthal, H. and Steiner, H.-G.: Chapter 13: Group Theory and Geometry, in Fundamentals of Mathematics, Vol. II: Geometry. (eds. H. Behnke e.a.), MIT Press, Cambridge, 516-533 (1974).
  • [25] Ganchev, G. and Mihova, V.: Riemannian manifolds of quasi-constant sectional curvature. J. reine angew. Math. 522, 119-141(2000).
  • [26] Ganchev, G. and Mihova, V.: A classification of Riemannian manifolds of quasi-constant sectional curvatures. Ann. Sofia Univ. Fac. Math. and Inf. 102, 115-138 (2015).
  • [27] Gunawan, H., Neswan, O. and Setya-Budhi, W.: A formule for angles between subspaces of inner product spaces. Beitr. Algebra Geom. 46, 311-320 (2009).
  • [28] Haesen, S. and Verstraelen, L.: Properties of a scalar curvature invariant depending on two planes. Manuscripta Math. 122, 59-72(2007).
  • [29] Haesen, S. and Verstraelen, L.: Natural Intrinsic Geometrical Symmetries. SIGMA 5, pp. 15 (2009). Special Issue ‘’Élie Cartan and Differential Geometry.
  • [30] Haesen, S. and Verstraelen, L.: Chapter 8: Curvatures and Symmetries of Parallel Transport, and Chapter 9: Extrinsic Symmetries of Parallel transport. in Differential Geometry and Topology, Discrete and Computational Geometry, (eds. J.-M. Morvan and M. Boucetta), NATO Science Series - IOS Press, 197-255(2005).
  • [31] Jahanara, B., Haesen, S., ¸Sentürk, Z. and Verstraelen, L.: On the parallel transport of the Ricci curvatures. J. Geom. Phys. 57, 1771-1777 (2007).
  • [32] Klein, F.: Elementary mathematics from an advanced standpoint - Geometry, Dover, New York (1939).
  • [33] Kühnel, W.: Differentialgeometrie. Kurven - Flächen - Mannigfaltigheiten. Vieweg, Wiesbaden (2008); English translation: Differential Geometry. Curves - Surfaces - Manifolds. AMS Student Mathematical Library 16 (2006).
  • [34] Kowalski, O. and Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three - elliptic case. Rend. Mat. Appl. 17(7), 477-512( 1997).
  • [35] Kowalski, O. and Sekizawa, M.: Pseudo-symmetric spaces of constant type in dimension three - non-elliptic case. Bull. Tokyo Gakugei Univ. 50(4), 1-28(1998).
  • [36] Levy, H.: Tensors determined by a hypersurface in Riemannian space. Trans. AMS 28, 671-694(1926).
  • [37] Lumiste, Ü.: Semi-parallel submanifolds in real space forms. Springer, Berlin (2009).
  • [38] Matveev, V.S.: Geometric explanation of the Beltrami theorem. Int. J. Methods Mod. Phys. 3, 623-629(2006).
  • [39] Mikeš, J., Kiosak, V. and Vanžurová, A.: Geodesic mappings of manifolds with affine connection. Palacký University, Olomouc (2008).
  • [40] Mikeš, J., Vanžurová, A. and Hinterleitner, I.: Geodesic mappings and some generalizations. Palacký University, Olomouc (2009).
  • [41] Risteki, I.B. and Trencevski, K.: Principal values and principal subspaces of two subspaces of vector spaces with inner product. Beitr. Algebra Geom. 42, 289-300(2001).
  • [42] Schouten, J.A.: Die direkte Analysis zur neueren Relativitätstheorie. Verhan. Konink. Akad. Wet. Amsterdam 12, nr. 6, 1-95(1918).
  • [43] Sekigawa, K. and Takagi, H.: On the conformally flat spaces satisfying a curvature condition on the Ricci tensor. Tôhoku Math. J. 23, 1-11 (1971).
  • [44] Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. I. The local version. J. Diff. Geom. 17 , 531-582 (1982).
  • [45] Szabó, Z.I.: Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0. II. The global version. Geom. Dedicata 19, 65-108(1985).
  • [46] Szabó, Z. I.: A short topological proof of the symmetry of 2-point homogeneous spaces. Invent. math. 106, 61-64 (1991).
  • [47] Thurston, W.: The Geometry and Topology of 3-Manifolds., Lecture Notes, Princeton University Press, Princeton (1979).
  • [48] Verstraelen, L.: Natural extrinsic geometrical symmetries - an introduction. AMS Contempory Mathematics, Vol. 674 5-16(2016). (from the ‘’AMS Special Session on Recent Advances in the Geometry of Submanifolds Dedicated to the Memory of Franki Dillen (1963-2013)”, (eds. B. Suceava e.a.).
  • [49] Verstraelen, L.: Comments on the pseudo-symmetry in the sense of Ryszard Deszcz. Geometry and Topology of Submanifolds VI, (eds. F. Dillen e.a.), World Scientific, Singapore, 199-209(1994).
  • [50] Verstraelen, L.: Submanifold theory - A contemplation of submanifolds, AMS Contemporary Mathematics, Vol. 756, 21-56(2020); (from the ‘’AMS Special Session on the Geometry of Submanifolds in Honor of Bang-Yen Chen”, (eds. J. Van der Veken e.a.)).
  • [51] Wall, C. T. C.: Geometries and Geometrical Structures in the real dimension 4 and the complex dimension 2. Springer Lecture Notes 1167, eds. Dold and Eckmann; Proc. Univ. of Maryland Special Year of Low Dimensional Topology , College Park, (1983-1984).
  • [52] Weyl, H.: Symmetry. Princeton University Press, Princeton (1952).
  • [53] Yano, K., Houh, C.-S. and Chen, B.-Y.: Intrinsic characterization of certain conformally flat spaces. Kodai Math. Sem. Rep. 25, 357-361(1973).
There are 53 citations in total.

Details

Primary Language English
Subjects Algebraic and Differential Geometry
Journal Section Research Article
Authors

Stefan Haesen This is me

Miroslava Petrović-torgašev

Leopold Verstraelen This is me

Early Pub Date April 7, 2024
Publication Date April 23, 2024
Submission Date March 15, 2024
Acceptance Date April 5, 2024
Published in Issue Year 2024 Volume: 17 Issue: 1

Cite

APA Haesen, S., Petrović-torgašev, M., & Verstraelen, L. (2024). On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms. International Electronic Journal of Geometry, 17(1), 232-244. https://doi.org/10.36890/iejg.1466330
AMA Haesen S, Petrović-torgašev M, Verstraelen L. On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms. Int. Electron. J. Geom. April 2024;17(1):232-244. doi:10.36890/iejg.1466330
Chicago Haesen, Stefan, Miroslava Petrović-torgašev, and Leopold Verstraelen. “On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms”. International Electronic Journal of Geometry 17, no. 1 (April 2024): 232-44. https://doi.org/10.36890/iejg.1466330.
EndNote Haesen S, Petrović-torgašev M, Verstraelen L (April 1, 2024) On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms. International Electronic Journal of Geometry 17 1 232–244.
IEEE S. Haesen, M. Petrović-torgašev, and L. Verstraelen, “On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms”, Int. Electron. J. Geom., vol. 17, no. 1, pp. 232–244, 2024, doi: 10.36890/iejg.1466330.
ISNAD Haesen, Stefan et al. “On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms”. International Electronic Journal of Geometry 17/1 (April 2024), 232-244. https://doi.org/10.36890/iejg.1466330.
JAMA Haesen S, Petrović-torgašev M, Verstraelen L. On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms. Int. Electron. J. Geom. 2024;17:232–244.
MLA Haesen, Stefan et al. “On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms”. International Electronic Journal of Geometry, vol. 17, no. 1, 2024, pp. 232-44, doi:10.36890/iejg.1466330.
Vancouver Haesen S, Petrović-torgašev M, Verstraelen L. On Thurston’s Geometrical Space Form Problem: On Quasi Space Forms. Int. Electron. J. Geom. 2024;17(1):232-44.
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