Geodesic
Theorems of Liuville types in theory mappings of the complete Riemannian manifolds
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 3, pp. 3-10

Voir la notice de l'article provenant de la source Math-Net.Ru

In the present paper we prove some vanishing theorems for projective and holomorphically projective diffeomorphisms of Riemannian and Kähler complete manifolds. We will use the well-known Liouville type theorems on harmonic, subharmonic and superharmonic functions on complete Riemannian manifolds for proofs of these theorems.
Keywords: complete Riemannian and Kähler manifolds, projective and holomorphically projective diffeomorphisms. 
I. A. Alexandrova; J. Mikeš; S. E. Stepanov; I. I. Tsyganok. Theorems of Liuville types in theory mappings of the complete Riemannian manifolds. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 15 (2015) no. 3, pp. 3-10. http://geodesic.mathdoc.fr/item/VNGU_2015_15_3_a0/
@article{VNGU_2015_15_3_a0,
     author = {I. A. Alexandrova and J. Mike\v{s} and S. E. Stepanov and I. I. Tsyganok},
     title = {Theorems of {Liuville} types in theory mappings of the complete {Riemannian} manifolds},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {3--10},
     year = {2015},
     volume = {15},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2015_15_3_a0/}
}
TY  - JOUR
AU  - I. A. Alexandrova
AU  - J. Mikeš
AU  - S. E. Stepanov
AU  - I. I. Tsyganok
TI  - Theorems of Liuville types in theory mappings of the complete Riemannian manifolds
JO  - Sibirskij žurnal čistoj i prikladnoj matematiki
PY  - 2015
SP  - 3
EP  - 10
VL  - 15
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VNGU_2015_15_3_a0/
LA  - ru
ID  - VNGU_2015_15_3_a0
ER  - 
%0 Journal Article
%A I. A. Alexandrova
%A J. Mikeš
%A S. E. Stepanov
%A I. I. Tsyganok
%T Theorems of Liuville types in theory mappings of the complete Riemannian manifolds
%J Sibirskij žurnal čistoj i prikladnoj matematiki
%D 2015
%P 3-10
%V 15
%N 3
%U http://geodesic.mathdoc.fr/item/VNGU_2015_15_3_a0/
%G ru
%F VNGU_2015_15_3_a0

[1] L. P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, 1926 | MR | Zbl

[2] N. S. Sinyukov, Geodesic Mappings of Riemannian Spaces, Nauka, M., 1979 (in Russian) | MR

[3] Mikeš J., Vanžurová A., Hiterleitner I., Geodesic Mappings and Some Generalizations, Palacký University Press, Olomouc, 2009 | MR | Zbl

[4] J. Mikeš, “Geodesic mappings of affine-connected and Riemannian spaces”, J. Math. Sci., 78:3 (1996), 311–333 | DOI | MR | Zbl

[5] J. Mikeš, “Holomorphically projective mappings and their generalizations”, J. Math. Sci., 89:3 (1998), 1334–1353 | DOI | MR | MR | Zbl

[6] Hiterleitner I., “On Holomorphically Projective Mappings of e-Kähler Manifolds”, Arch. Math. Brno, 48 (2012), 333–338 | DOI | MR

[7] Matveev V. S., “Geodesically Equivalent Metric in General Relativity”, J. Geom. Phys., 62:3 (2012), 675–691 | DOI | MR | Zbl

[8] Mikeš J., Stepanova E. S., Tsyganok I. I., “A Geodesic Mapping and its Field of Symmetric Linear Endomorphisms”, Differ. Geom. Appl., 35 (2014), 44–49 | DOI | MR | Zbl

[9] E. N. Sinyukova, “Geodesic mappings of certain special Riemannian spaces”, Math. Notes, 30 (1982), 946–949 | DOI | MR | Zbl

[10] N. S. Sinyukov, E. N. Sinyukova, “Holomorphically projective mappings of special Kähler spaces”, Math. Notes, 36 (1984), 706–709 | DOI | MR | Zbl

[11] Stepanov S. E., “On the Global Theory of Some Classes of Mappings”, Ann. Global Anal. Geom., 13:3 (1995), 239–249 | DOI | MR | Zbl

[12] Hiterleitner I., “Geodesic Mappings on Compact Riemannian Manifolds with Conditions on Sectional Curvature”, Publ. Inst. Math. (Beograd) (N.S.), 94:108 (2013), 125–130 | DOI | MR

[13] Ni L., “Vanishing Theorems on Complete Kähler Manifolds and Their Applications”, J. Diff. Geom., 50 (1998), 89–122 | MR

[14] S. E. Stepanov, “Vanishing theorems in affine, Riemannian, and Lorentz geometries”, J. Math. Sci., 141:1 (2007), 929–964 | DOI | MR | Zbl

[15] Vilms J., “Totally Geodesic Maps”, J. Differ. Geom., 4 (1970), 73–79 | MR | Zbl

[16] Sh. Kobayashi, K. Nomizu, Foundations of Differential Geometry, v. I, Interscience Publishers, New York–London, 1963 | MR | Zbl

[17] Yau S.-T., “Some Function-Theoretic Properties of Complete Riemannian Manifold and Their Applications to Geometry”, Indiana Univ. Math. J., 25 (1976), 659–670 | DOI | MR | Zbl

[18] Jorgenson J., Walling L. (eds.), The Ubiquitous Heat Kernel, AMS, Providence, 2006 | MR | Zbl

[19] Ichihara K., “Curvature, Geodesics and the Brownian Motion on a Riemannian Manifold. I: Recurrence Properties”, Nagoya Math. J., 87 (1982), 101–114 | MR | Zbl

[20] Ichihara K., “Curvature, Geodesics and the Brownian Motion on a Riemannian Manifold. II: Explosion properties”, Nagoya Math. J., 87 (1982), 115–125 | MR | Zbl

[21] I. A. Aleksandrova, “The spectral method in asymptotic diffusion problems with drift”, Math. Notes, 59:5 (1996), 554–556 | DOI | DOI | MR | Zbl

[22] A. A. Grigor'yan, “Stochastically complete manifolds and summable harmonic functions”, Math. USSR, Izv., 33:2 (1989), 425–432 | DOI | MR | Zbl | Zbl

[23] Grigor'yan A. A., “Heat Kernel and Analysis on Manifolds”, The Ubiquitous Heat Kernel, AMS Special Session (Boulder, CO, USA, October 2–4, 2003), AMS/IP Studies in Adv. Math., AMS, Providence, 2009, 93–191 | MR

[24] Hsu P., “Heat Semigroup on a Complete Riemannian Manifold”, Ann. Probab., 17:3 (1989), 1248–1254 | DOI | MR | Zbl

[25] Yau S.-T., “On the Heat Kernel of Complete Riemannian Manifold”, J. Math. Pures Appl., 9:57 (1978), 191–201 | MR | Zbl

[26] Azencott R., “Behavior of Diffusion Semi-Groups at Infinity”, Bull. Soc. Math. Fr., 102 (1974), 193–240 | MR | Zbl

[27] Grigor'yan A., “Analytic and Geometric Background of Recurrence and Non-Explosion of the Brownian Motion on Riemannian Manifolds”, Bull. Am. Math. Soc., New Ser., 36:2 (1999), 135–249 | DOI | MR | Zbl

[28] Cheng S. Y., Yau S.-T., “Differential Equations on Riemannian Manifolds and their Geometrical Applications”, Commun. Pure Appl. Math., 28 (1975), 333–354 | DOI | MR | Zbl

[29] Yano K., “Sur la Correspondence Projective Entre Deux Espaces Pseudo-Hermitens”, C. R. Acad. Sci., Paris, Sér. A, 239 (1954), 1346–1348 (in French) | MR | Zbl

[30] Otsuki T., Tashiro Y., “On Curves in Kaehlerian Spaces”, Math. J. Okayama Univ., 4 (1954), 57–78 | MR | Zbl

[31] Tashiro Y., “On a Holomorphically Projective Correspondence in an Almost Complex Space”, Math. J. Okayama Univ., 6 (1957), 147–152 | MR | Zbl