Geodesic
Shells of monotone curves
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 677-699
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

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.
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.
DOI : 10.1007/s10587-015-0202-5
Classification : 51H20, 53B05, 53B20, 53B30, 53C22
Keywords: geodesic; shell of a curve; affine connection; (pseudo-)Riemannian metric; projective equivalence 
@article{10_1007_s10587_015_0202_5,
     author = {Mike\v{s}, Josef and Strambach, Karl},
     title = {Shells of monotone curves},
     journal = {Czechoslovak Mathematical Journal},
     pages = {677--699},
     year = {2015},
     volume = {65},
     number = {3},
     doi = {10.1007/s10587-015-0202-5},
     mrnumber = {3407599},
     zbl = {06537686},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0202-5/}
}
TY  - JOUR
AU  - Mikeš, Josef
AU  - Strambach, Karl
TI  - Shells of monotone curves
JO  - Czechoslovak Mathematical Journal
PY  - 2015
SP  - 677
EP  - 699
VL  - 65
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0202-5/
DO  - 10.1007/s10587-015-0202-5
LA  - en
ID  - 10_1007_s10587_015_0202_5
ER  - 
%0 Journal Article
%A Mikeš, Josef
%A Strambach, Karl
%T Shells of monotone curves
%J Czechoslovak Mathematical Journal
%D 2015
%P 677-699
%V 65
%N 3
%U http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0202-5/
%R 10.1007/s10587-015-0202-5
%G en
%F 10_1007_s10587_015_0202_5
Mikeš, Josef; Strambach, Karl. Shells of monotone curves. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 3, pp. 677-699. doi: 10.1007/s10587-015-0202-5

[1] Betten, D.: Topological shift spaces. Adv. Geom. 5 (2005), 107-118. | DOI | MR | Zbl

[2] Betten, D.: Some classes of topological 3-spaces. German Result. Math. 12 (1987), 37-61. | MR | Zbl

[3] Cartan, E.: Les espaces riemanniens symétriques. French Verh. Internat. Math.-Kongr. 1 (1932), 152-161. | Zbl

[4] Eisenhart, L. P.: Non-Riemannian geometry. American Mathematical Society Colloquium Publications 8 American Mathematical Society, Providence (1990). | MR

[5] Gerlich, G.: Topological affine planes with affine connections. Adv. Geom. 5 (2005), 265-278. | DOI | MR | Zbl

[6] Gerlich, G.: Stable projective planes with Riemannian metrics. Arch. Math. 79 (2002), 317-320. | DOI | MR | Zbl

[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). | MR | Zbl

[9] Kagan, V. F.: Subprojective Spaces. Russian Bibliothek der Russischen Wissenschaften Mathematik, Mechanik, Physik, Astronomie Staatsverlag für Physikalisch-Mathematische Literatur, Moskva (1961). | MR

[10] Kamke, E.: Differentialgleichungen. Lösungsmethoden und Lösungen. 1. Gewöhnliche Differentialgleichungen. German Akademische Verlagsgesellschaft, Leipzig (1942). | Zbl

[11] Kamke, E.: Differentialgleichungen Reeller Funktionen. Akademische Verlagsgesellschaft, Leipzig (1930), German.

[12] Mikeš, J.: Geodesic mappings of affine-connected and Riemannian spaces. J. Math. Sci., New York 78 (1996), 311-333. | DOI | MR

[13] Mikeš, J., Strambach, K.: Grünwald shift spaces. Publ. Math. 83 (2013), 85-96. | MR | Zbl

[14] Mikeš, J., Strambach, K.: Differentiable structures on elementary geometries. Result. Math. 53 (2009), 153-172. | DOI | MR

[15] Mikeš, J., Vanžurová, A., Hinterleitner, I.: Geodesic Mappings and Some Generalizations. Palacký University, Faculty of Science, Olomouc (2009). | MR | Zbl

[16] Norden, A. P.: Spaces with Affine Connection. Russian Nauka Moskva (1976). | MR | Zbl

[17] Petrov, A. Z.: New Methods in the General Theory of Relativity. Russian Hauptredaktion für Physikalisch-Mathematische Literatur Nauka, Moskva (1966). | MR

[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). | MR

[19] Sinyukov, N. S.: Geodesic Mappings of Riemannian Spaces. Russian Nauka Moskva (1979). | MR | Zbl

[20] Yano, K., Bochner, S.: Curvature and Betti Numbers. Annals of Mathematics Studies 32 Princeton University Press 9, Princeton (1953). | MR | Zbl

Cité par Sources :