Abstract:
In this survey, we consider one aspect of the Bochner technique, the proof of vanishing theorems by using the Weitzenbock integral formulas, which allows us to extend the technique to pseudo-Riemannian manifolds and equiaffine connection manifolds.
Citation:
S. E. Stepanov, “Vanishing theorems in affine, Riemannian, and Lorenz geometries”, Fundam. Prikl. Mat., 11:1 (2005), 35–84; J. Math. Sci., 141:1 (2007), 929–964
This publication is cited in the following 19 articles:
Vladimir Rovenski, Sergey Stepanov, Irina Tsyganok, Springer Proceedings in Mathematics & Statistics, 440, Differential Geometric Structures and Applications, 2024, 167
S. E. Stepanov, I. Mikesh, I. I. Tsyganok, “Obobschennaya tekhnika Bokhnera i ee primenenie k izucheniyu proektivnykh i konformnykh otobrazhenii”, Materialy Mezhdunarodnoi konferentsii «Klassicheskaya i sovremennaya geometriya», posvyaschennoi 100-letiyu so dnya rozhdeniya professora Levona Sergeevicha Atanasyana (15 iyulya 1921 g.—5 iyulya 1998 g.). Moskva, 1–4 noyabrya 2021 g. Chast 4, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 223, VINITI RAN, M., 2023, 112–122
Josef Mikeš, Lenka Rýparová, Sergey Stepanov, Irina Tsyganok, “On the Geometry in the Large of Einstein-like Manifolds”, Mathematics, 10:13 (2022), 2208
I. Mikesh, S. Formella, I. Ginterleitner, N. I. Guseva, “Nekotorye voprosy geodezicheskikh otobrazhenii prostranstv Einshteina”, Geometriya, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 203, VINITI RAN, M., 2021, 50–61
Vladimir Rovenski, Sergey Stepanov, Irina Tsyganok, “A Generalized Bochner Technique and Its Application to the Study of Conformal Mappings”, Axioms, 10:4 (2021), 333
Josef Mikeš, Vladimir Rovenski, Sergey E. Stepanov, “An example of Lichnerowicz-type Laplacian”, Ann Glob Anal Geom, 58:1 (2020), 19
Sergey Stepanov, Irina Tsyganok, “Vanishing theorems for harmonic mappings into non-negatively curved manifolds and their applications”, manuscripta math., 154:1-2 (2017), 79
D. M. Volobuev, N. G. Makarenko, I. S. Knyazeva, “Evolution of the Bochner Laplacian for magnetograms of solar active regions”, Geomagn. Aeron., 56:7 (2016), 886
I. A. Alexandrova, J. Mikeš, S. E. Stepanov, I. I. Tsyganok, “Theorems of Liuville types in theory mappings of the complete Riemannian manifolds”, J. Math. Sci., 221:6 (2017), 737–744
S. E. Stepanov, I. A. Alexandrova, I. I. Tsyganok, J. Mikeš, “Conformal Killing forms on totally umbilical submanifolds”, Journal of Mathematical Sciences, 217:5 (2016), 525–539
Sergey E. Stepanov, Josef Mikeš, “The generalized Landau–Raychaudhuri equation and its applications”, Int. J. Geom. Methods Mod. Phys., 12:08 (2015), 1560026
Sergey E. Stepanov, Marek Jukl, Josef Mikeš, Springer Proceedings in Mathematics & Statistics, 85, Algebra, Geometry and Mathematical Physics, 2014, 495
Sergey E. Stepanov, Marek Jukl, Josef Mikeš, “Vanishing theorems of conformal Killing forms and their applications to electrodynamics in the general relativity theory”, Int. J. Geom. Methods Mod. Phys., 11:09 (2014), 1450039
Sergey E. Stepanov, Josef Mikeš, “Betti and Tachibana numbers of compact Riemannian manifolds”, Differential Geometry and its Applications, 31:4 (2013), 486