Shells of monotone curves

Josef Mikeš; Karl Strambach

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 3, page 677-699
  • ISSN: 0011-4642

Abstract

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We determine in ℝ n the form of curves C corresponding to strictly monotone functions as well as the components of affine connections ∇ for which any image of C under a compact-free group Ω of affinities containing the translation group is a geodesic with respect to ∇ . Special attention is paid to the case that Ω contains many dilatations or that C is a curve in ℝ 3 . If C is a curve in ℝ 3 and Ω is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when ∇ yields a flat or metrizable space and compute the corresponding metric tensor.

How to cite

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Mikeš, Josef, and Strambach, Karl. "Shells of monotone curves." Czechoslovak Mathematical Journal 65.3 (2015): 677-699. <http://eudml.org/doc/271796>.

@article{Mikeš2015,
abstract = {We determine in $\mathbb \{R\}^n$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla $ for which any image of $C$ under a compact-free group $\Omega $ of affinities containing the translation group is a geodesic with respect to $\nabla $. Special attention is paid to the case that $\Omega $ contains many dilatations or that $C$ is a curve in $\mathbb \{R\}^3$. If $C$ is a curve in $\mathbb \{R\}^3$ and $\Omega $ is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when $\nabla $ yields a flat or metrizable space and compute the corresponding metric tensor.},
author = {Mikeš, Josef, Strambach, Karl},
journal = {Czechoslovak Mathematical Journal},
keywords = {geodesic; shell of a curve; affine connection; (pseudo-)Riemannian metric; projective equivalence},
language = {eng},
number = {3},
pages = {677-699},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Shells of monotone curves},
url = {http://eudml.org/doc/271796},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Mikeš, Josef
AU - Strambach, Karl
TI - Shells of monotone curves
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 3
SP - 677
EP - 699
AB - We determine in $\mathbb {R}^n$ the form of curves $C$ corresponding to strictly monotone functions as well as the components of affine connections $\nabla $ for which any image of $C$ under a compact-free group $\Omega $ of affinities containing the translation group is a geodesic with respect to $\nabla $. Special attention is paid to the case that $\Omega $ contains many dilatations or that $C$ is a curve in $\mathbb {R}^3$. If $C$ is a curve in $\mathbb {R}^3$ and $\Omega $ is the translation group then we calculate not only the components of the curvature and the Weyl tensor but we also decide when $\nabla $ yields a flat or metrizable space and compute the corresponding metric tensor.
LA - eng
KW - geodesic; shell of a curve; affine connection; (pseudo-)Riemannian metric; projective equivalence
UR - http://eudml.org/doc/271796
ER -

References

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  1. Betten, D., 10.1515/advg.2005.5.1.107, Adv. Geom. 5 (2005), 107-118. (2005) Zbl1067.51007MR2110464DOI10.1515/advg.2005.5.1.107
  2. Betten, D., Some classes of topological 3-spaces, German Result. Math. 12 (1987), 37-61. (1987) Zbl0631.51006MR0911460
  3. Cartan, E., Les espaces riemanniens symétriques, French Verh. Internat. Math.-Kongr. 1 (1932), 152-161. (1932) Zbl0006.42102
  4. Eisenhart, L. P., Non-Riemannian geometry, American Mathematical Society Colloquium Publications 8 American Mathematical Society, Providence (1990). (1990) MR1466961
  5. Gerlich, G., 10.1515/advg.2005.5.2.265, Adv. Geom. 5 (2005), 265-278. (2005) Zbl1080.51007MR2131819DOI10.1515/advg.2005.5.2.265
  6. Gerlich, G., 10.1007/s00013-002-8318-x, Arch. Math. 79 (2002), 317-320. (2002) Zbl1022.51012MR1944956DOI10.1007/s00013-002-8318-x
  7. Hinterleitner, I., Mikeš, J., Geodesic mappings onto Weyl spaces, Proc. 8th Int. Conf. on Appl. Math (APLIMAT 2009) Bratislava 423-430. 
  8. Irving, R. S., Integers, Polynomials, and Rings. A Course in Algebra, Undergraduate Texts in Mathematics Springer, New York (2004). (2004) Zbl1046.00002MR2025456
  9. Kagan, V. F., Subprojective Spaces, Russian Bibliothek der Russischen Wissenschaften Mathematik, Mechanik, Physik, Astronomie Staatsverlag für Physikalisch-Mathematische Literatur, Moskva (1961). (1961) MR0131242
  10. Kamke, E., Differentialgleichungen. Lösungsmethoden und Lösungen. 1. Gewöhnliche Differentialgleichungen, German Akademische Verlagsgesellschaft, Leipzig (1942). (1942) Zbl0026.31801
  11. Kamke, E., Differentialgleichungen Reeller Funktionen, Akademische Verlagsgesellschaft, Leipzig (1930), German. (1930) 
  12. Mikeš, J., 10.1007/BF02365193, J. Math. Sci., New York 78 (1996), 311-333. (1996) MR1384327DOI10.1007/BF02365193
  13. Mikeš, J., Strambach, K., Grünwald shift spaces, Publ. Math. 83 (2013), 85-96. (2013) Zbl1289.51008MR3081228
  14. Mikeš, J., Strambach, K., 10.1007/s00025-008-0296-2, Result. Math. 53 (2009), 153-172. (2009) MR2481410DOI10.1007/s00025-008-0296-2
  15. Mikeš, J., Vanžurová, A., Hinterleitner, I., Geodesic Mappings and Some Generalizations, Palacký University, Faculty of Science, Olomouc (2009). (2009) Zbl1222.53002MR2682926
  16. Norden, A. P., Spaces with Affine Connection, Russian Nauka Moskva (1976). (1976) Zbl0925.53007MR0467565
  17. Petrov, A. Z., New Methods in the General Theory of Relativity, Russian Hauptredaktion für Physikalisch-Mathematische Literatur Nauka, Moskva (1966). (1966) MR0207365
  18. Salzmann, H., Betten, D., Grundhöfer, T., Hähl, H., Löwen, R., Stroppel, M., Compact Projective Planes. With an Introduction to Octonion Geometry, De Gruyter Expositions in Mathematics 21 De Gruyter, Berlin (1995). (1995) MR1384300
  19. Sinyukov, N. S., Geodesic Mappings of Riemannian Spaces, Russian Nauka Moskva (1979). (1979) Zbl0637.53020MR0552022
  20. Yano, K., Bochner, S., Curvature and Betti Numbers, Annals of Mathematics Studies 32 Princeton University Press 9, Princeton (1953). (1953) Zbl0051.39402MR0062505

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