On an application of the Stokes' theorem in global Riemannian geometry

S. E. Stepanov

Fundamentalʹnaâ i prikladnaâ matematika, Tome 8 (2002) no. 1, pp. 245-262.

Résumé

Applying the Stokes' theorem we have deduced the Weitzenbock's formula for symmetric 2-forms on a compact Riemannian manifold $M$ with boundary $\partial M\neq\varnothing$. Using the formula we have proved that Killing symmetric 2-forms and Killing $p$-forms on a Riemannian manifold $M$ of non-positive sectional curvature and convex boundary $\partial M$ must be either parallel or zero. Finally, we have applied our results to the global theory of projective and umbilical maps.

Bibliographie

[1] Besse A., Mnogoobraziya Einshteina, T. 1, 2, Mir, M., 1990 | MR | Zbl

[2] Yano K., Bokhner S., Krivizna i chisla Betti, IL, M., 1957

[3] Yano K., Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970 | MR | Zbl

[4] Wu H., “The Bochner technique”, Proc. Beijing Symp. Differ. Geom. and Differ. Equat., V. 2 (Aug. 18–Sept. 21, 1980), Science Press Gordon–Breach, New York, 1982, 929–1071 | MR

[5] Kramer D. i dr., Tochnye resheniya uravneniya Einshteina, Energoizdat, M., 1982

[6] Burginon Zh.-P., “Formuly Veitsenbëka v razmernosti 4”, Chetyrëkhmernaya rimanova geometriya, Mir, M., 1985, 260–279 | MR

[7] Stepanov S. E., “The seven classes of almost symplectic structures”, Webs Quasigroups, Tver' State University, Tver', 1992, 93–96 | MR | Zbl

[8] Stepanov S. E., “A class of closed forms and special Maxwell equations”, Proc. Conference on Differential Geometry (Budapest, July 27–30, 1996), Budapest, 1996, 113

[9] Kashiwada T., “On conformal Killing tensor”, Natural Science Report, Ochanomizu University, 19:2 (1968), 67–74 | MR | Zbl

[10] Tashibana Sh., “On conformal Killing tensor in a Riemannian space”, Tohoku Math. Journ., 21 (1969), 56–64 | DOI | MR

[11] Stepanov S. E., “O primenenii odnoi teoremy P. A. Shirokova v tekhnike Bokhnera”, Izv. vyssh. uchebn. zaved. Matematika, 1996, no. 9, 53–59 | MR | Zbl

[12] Stepanov S. E., “Formy Killinga na kompaktnom mnogoobrazii s kraem”, Sovremennaya geometriya i teoriya fizicheskikh polei, Tezisy dokladov Mezhdunarodnogo geometricheskogo seminara im. N. I. Lobachevskogo (Kazan, 4–6 fevralya 1997), Kazan, 1997, 114

[13] Stepanov S. E., “On the global theory of some classes of mappings”, Annals of Global Analysis and Geometry, 13:3 (1995), 239–249 | DOI | MR | Zbl

[14] Narasimkhan R., Analiz na deistvitelnykh i kompleksnykh mnogoobraziyakh, Mir, M., 1971 | Zbl

[15] Zulanke R., Vintgen P., Differentsialnaya geometriya i rassloeniya, Mir, M., 1975

[16] Gromol D. i dr., Rimanova geometriya v tselom, Mir, M., 1971

[17] Stepanov S. E., “Simmetricheskie tenzory na kompaktnom rimanovom mnogoobrazii”, Mat. zametki, 52:4 (1992), 85–88 | MR | Zbl

[18] Shirokov P. A., “Postoyannye polya vektorov i tenzorov vtorogo poryadka v Riemann'ovykh prostranstvakh”, Izv. fiz.-matem. o-va, 25 (1925), 86–114, Kazan

[19] Yamaguchi S., “On a theorem of Gallot–Meyer–Tachibana in Riemannian manifolds of positive curvature operator”, TRU Math., 11 (1975), 17–22 | MR | Zbl

[20] Eizenkhart L. P., Rimanova geometriya, IL, M., 1948

[21] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, T. 2, Nauka, M., 1981

[22] Davidov I., Sergeev A. G., “Tvistornye prostranstva i garmonicheskie otobrazheniya”, Uspekhi mat. nauk, 48:3 (1993), 3–96 | MR | Zbl

[23] Har' El Zvi, “Projective mappings and distortion theorems”, J. Differential Geometry, 15 (1980), 97–106 | MR | Zbl

[24] Nore T., “Second fundamental form of a map”, Ann. Mat. Pure et Appl., 146 (1987), 281–310 | DOI | MR | Zbl

[25] Yano K., Ishihata Sh., “Harmonic and relatively affine mappings”, J. Differential Geometry, 10 (1975), 501–509 | MR | Zbl

[26] Sinyukov N. S., Geodezicheskie otobrazheniya rimanovykh prostranstv, Nauka, M., 1979 | MR | Zbl

[27] Mikeš J., “Global geodesic mappings and their generalization for compact Riemannian space”, Proc. Conf. on Diff. Geom. and its Appl. (Opava, August 24–28, 1992), Opava, 1992, 143–149 | MR