# zbMATH — the first resource for mathematics

Extremal Lorentzian surfaces with null $$r$$-planar geodesics in space forms. (English) Zbl 1334.53057
Summary: We show a congruence theorem for oriented Lorentzian surfaces with horizontal reflector lifts in pseudo-Riemannian space forms of neutral signature. As a corollary, a characterization theorem is obtained for the Lorentzian Boruvka spheres, that is, a full real analytic null $$r$$-planar geodesic immersion with vanishing mean curvature vector field is locally congruent to the Lorentzian Boruvka sphere in a $$2r$$-dimensional space form of neutral signature.

##### MSC:
 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
Full Text:
##### References:
 [1] R. L. Bryant, Minimal surfaces of constant curvature in $$S^n$$, Trans. Amer. Math. Soc. 290 (1985), 259-271. · Zbl 0572.53002 · doi:10.2307/1999793 [2] E. Calabi, Minimal immersions of surfaces in Euclidean spheres, J. Differential Geometry 1 (1967), 111-125. · Zbl 0171.20504 · doi:10.4310/jdg/1214427884 [3] B. Y. Chen, Classification of minimal Lorentz surfaces in indefinite space forms with arbitrary codimension and arbitrary index, Publ. Math. Debrecen 78 (2011), no. 2, 485-503. · Zbl 1274.53078 · doi:10.5486/PMD.2011.4860 [4] S. S. Chern, On the minimal immersions of the two-sphere in a space of constant curvature, Problems in Analysis, pp. 27-40, University Press, Princeton, 1970. · Zbl 0217.47601 [5] Q. Ding and J. Wang, A remark on harmonic maps between pseudo-Riemannian spheres, Northeast. Math. J. 12 (1996), 217-221. · Zbl 0863.58019 [6] G. Jensen and M. Rigoli, Neutral surfaces in neutral four-spaces, Matematiche (Catania) 45 (1990), 407-443. · Zbl 0757.53035 [7] Y. H. Kim, Surfaces of a Euclidean space with helical or planar geodesics through a point, Ann. Mat. Pura Appl. (4) 164 (1993), 1-35. · Zbl 0791.53010 · doi:10.1007/BF01759312 [8] K. Miura, Helical geodesic immersions of semi-Riemannian manifolds, Kodai Math. J. 30 (2007), 322-343. · Zbl 1138.53045 · doi:10.2996/kmj/1193924937 [9] K. Miura, Construction of harmonic maps between semi-Riemannian spheres, Tsukuba J. Math. 31 (2007), 397-409. · Zbl 1138.53054 [10] K. Miura, Isometric immersions with geodesic normal sections in semi-Riemannian geometry, Tokyo J. Math. 31 (2008), 479-488. · Zbl 1171.53038 · doi:10.3836/tjm/1233844064 [11] B. O’Neill, Semi-Riemannian geometry, Academic Press Inc., New York, 1983. [12] M. Sakaki, Two classes of Lorentzian stationary surfaces in semi-Riemannian space forms, Nihonkai Math. J. 15 (2004), 15-22. · Zbl 1063.53075 [13] K. Sakamoto, Helical immersions into a unit sphere, Math. Ann. 261 (1982), 63-80. · Zbl 0476.53033 · doi:10.1007/BF01456411 · eudml:182866 [14] M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish Inc., 1975. · Zbl 0306.53002 [15] N. R. Wallach, Extension of locally defined minimal immersions into spheres, Arch. Math. (Basel) 21 (1970), 210-213. · Zbl 0193.22001 · doi:10.1007/BF01220905
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.