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.
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References
Affine Differentialgeometrie, 48, Oberwolfach (1986), pp. 1–24.
M. A. Akivis, Higher-Dimensional Differential Geometry [in Russian], Kalinin University, Kalinin (1977).
K. Akutagawa, “On spacelike hypersurfaces with constant mean curvature in the de Sitter space,” Math. Z., 196, 13–19 (1987).
J. A. Aledo and L. J. Alias, “Curvature properties of compact spacelike hypersurfaces in de Sitter space,” Differ. Geom. Appl., 14, No. 2, 137–149 (2001).
V. D. Alekseevskii, A. M. Vinogradov, and V. V. Lychagin, “Main notions of differential geometry,” In: Contemporary Problems of Mathematics. Fundamental Directions, 28, All-Union Institute for Scientific and Technical Information, Moscow (1988), pp.5–289.
L. J. Alias and J. A. Pastor, “Spacelike hypersurfaces with constant scalar curvature in the Lorentz-Minkowski space,” Ann. Global Anal. Geom., 18, 75–83 (2000).
A. V. Aminova, “Transformation groups of Riemannian manifolds,” in: Problems in Geometry, 22, All-Union Institute for Scientific and Technical Information, Moscow (1990), pp. 97–165.
H. Bahn and S. Hong, “Geometric inequalities for spacelike hypersurfaces in the Minkowski spacetime,” Geom. Phys., 37, 94–99 (2001).
D. Baleanu and S. Codoban, “Killing tensor and separable coordinates in (1+1)-dimensions,” Rom. J. Phys., 44, Nos. 9–10, 933–938 (1999).
J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry, Pure Appl. Math., 67, Marcel Dekker, New York-Basel (1981).
J. K. Beem, P. E. Ehrlich, and S. Markvorsen, “Time-like isometries of space-times with nonnegative sectional curvature,” in: Top. Diff. Geom.: Colloq. Math. Soc. J. Bolyai, Debrecen, Aug. 26–Sept. 1, 1984, 1, Amsterdam (1988), pp. 153–165.
M. Bektash and M. Ergut, “Compact spacetime hypersurfaces in the de Sitter space,” Proc. Inst. Math. Mech. Azerbaijan, 10, 20–24 (1999).
L. Bérard, M. Berger, and C. Houzel, eds., Géométrie Riemannienne en Dimension 4. Seminaire Arthur Besse 1978/79 [in French], Text. Math., 3, CEDIC/Fernand Nathan. Paris (1981).
P. H. Berard, “From vanishing theorem to estimating theorem: the Bochner technique revisited,” Bull. Amer. Math. Soc., 19, No. 2, 371–402 (1988).
P. H. Berard, “A note on Bochner type theorems for complete manifolds,” Manuscr. Math., 69, No. 3, 261–266 (1990).
A. L. Besse, Einstein Manifolds, Springer-Verlag, Berlin (1987).
R. L. Bishop and B. O’Neill, “Manifolds of negative curvature,” Trans. Amer. Math. Soc., 145, 1–49 (1969).
G. Bitis, “Riemannian manifolds which admit a unique harmonic or Killing tensor field,” Tensor, N.S., 48, No. 1, 1–10 (1989).
G. Bitis, “Harmonic forms and Killing tensor fields,” Tensor, N.S., 55, No. 3, 215–222 (1994).
G. Bitis and G. Tsagas, “On the harmonic and Killing tensor field on a compact Riemannian manifolds,” Balkan J. Geom. Appl., 6, No. 2, 99–108 (2001).
W. Blashke and K. Reidemeister, Vorlesungen über Differential Geometrie. Bd. II. Affine Differential Geometrie, Springer-Verlag, Berlin (1923).
M. Blau, “Symmetries and pseudo-Riemannian manifold,” Rep. Math. Phys., 25, No. 1, 109–116 (1988).
S. Bochner, “Vector fields and Ricci curvature,” Bull. Amer. Math. Soc., 52, 776–797 (1946).
S. Bochner and K. Yano, Curvature and Betti Numbers, Princeton Univ. Press, Princeton (1953).
B. W. Brock and J. M. Steinke, “Local restrictions on nonpositively curved n-manifolds in ℝn+p,” Pac. J. Math., 196, No. 2, 271–281 (2000).
B.-Y. Chen and T. Nagano, “Harmonic metric, harmonic tensors, and Gauss maps,” J. Math. Soc. Jpn., 36, No. 2, 295–313 (1984).
S. S. Chern, “The geometry of G-structure,” Bull. Amer. Math. Soc., 72, 167–219 (1966).
Y. Choquet-Bruhat, “Mathematical problems in general relativity,” Usp. Mat. Nauk, 40, No. 6, 3–39 (1985).
C. D. Colinson, “The existence of Killing tensors in empty space-times,” Tensor, N.S., 28, 173–176 (1974).
C. D. Collinson and L. Howarth, “Generalized Killing tensor,” Gen. Relativ. Gravit., 32, No. 9, 1767–1776 (2000).
W. Dietz and R. Rudiger, “Space-times admitting Killing-Yano tensor, I,” Proc Roy. Soc. London, Ser. A., 375, 361–378 (1981).
W. Dietz and R. Rudiger, “Space-times admitting Killing-Yano tensor, II,” Proc Roy. Soc. London, Ser. A., 381, 315–322 (1982).
G. F. D. Duff and D. C. Spencer, “Harmonic tensor on Riemannian with bundary,” Ann. Math., 56, No. 1, 128–156 (1952).
M. P. Dussan and M. H. Noronha, “Manifolds with 2-nonnegative Ricci operator,” Pac. J. Math., 2, 319–334 (2002).
L. P. Eisenhart, Riemannian Geometry, Princeton Univ. Press, Oxford University Press, Princeton, New Jersey, London (1967).
H. V. Fagundes, “Closed spaces in cosmology,” Gen. Relativ. Gravit., 24, No. 2, 199–217 (1992).
C. M. Fulton, “Parallel vector fields,” Proc. Amer. Math. Soc., 16, 136–137 (1965).
G. J. Galloway, “Some global aspect of compact space-time,” Arch. Math., 42, No. 2, 168–172 (1984).
G. Ganchev and S. Ivanov, “Harmonic and holomorphic 1-forms on compact balanced Hermitian manifold,” Differ. Geom. Appl., 14, No. 1, 79–93 (2001).
A. Gray and L. Hervella, “The sixteen classes of almost Hermitean manifolds,” Ann. Math. Pura Appl., 123, 35–58 (1980).
D. Gromoll, W. Klingenberg, and W. Meyer, Riemannsche Geometrie im Großen, Lect. Notes Math., 55, Springer-Verlag, Berlin-Heidelberg-New York (1975).
M. Gromov, The Sign and Geometric Meaning of the Curvature [Russian translation], Izhevsk (1999).
K. Grotemeyer, “Die Integralsätze der affinen Flächentheorie,” Arch. Math., 3, 38–43 (1952).
R. S. Hamilton, “Four-manifolds with positive curvature operator,” J. Differ. Geom., 24, 153–179 (1986).
S. G. Harris, “What is the shape of space in spacetime?” in: Proc. Summer Res. Inst. Differ. Geom., Los Angeles, July 8–28, 1990, Providence, Rhode Island (1993), pp. 287–296.
S. W. Hawking and G. F. R. Ellis, The Large Scale Structure of Space-Time, Cambridge Monogr. Math. Phys., 1, Cambridge Univ. Press, London (1973).
S. Hawking and R. Penrose, The Nature of Space and Time, Princeton Univ. Press, Princeton, New Jersey (1996).
V. M. Isaev and S. E. Stepanov, “Examples of Killing and conformal Killing forms,” Diff. Geom. Mnogoobr. Figur, 32, 52–57 (2001).
Sh. Ishihara, “The integral formulas and their applications in some affinely connected manifolds,” Kodai Math. Semin. Repts., 13, No. 2, 93–108 (1961).
J.-B. Jun, Sh. Ayabe, and S. Yamaguchi, “On conformal Killing p-form in compact Kählerian manifolds,” Tensor, N.S., 42, No. 3, 258–271 (1985).
J.-B. Jun and S. Yamaguchi, “On projective Killing p-forms in Riemannian manifolds,” Tensor, N.S., 43, 157–166 (1986).
J. Kalina, B. Orsted, A. Pierzchalski, P. Walczak, F. Zhang, “Elliptic gradients and highest weights,” Bull. Acad. Polon. Sci., Ser. Math., 44, 511–519 (1996).
J. Kalina, A. Pierzchalski, and P. Walczak, “Only one of generalized gradients can be elliptic,” Ann. Polon. Math., 67, No. 2, 111–120 (1997).
T. Kashiwada, “On conformal Killing tensor,” Nat. Sci. Rep. Ochanomizu Univ., 19, 67–74 (1968).
A. Yu. Khokhlov, “On the maximum principle in the sense of L p , Dokl. Ross. Akad. Nauk, 348, No. 4, 452–454 (1996).
W. P. A. Klingeberg, “Affine differential geometry, by Katsumi Nomizu and Takeshi Sasaki. Book reviews,” Bull. Amer. Math. Soc., 33, No. 1, 75–76 (1996).
V. V. Klishevich, “Exact solution of Dirac and Klein-Gordon-Fock equations in a curved space admitting a second Dirac operator,” Class. Quantum Grav., 18, 3735–3752 (2001).
V. V. Klishevich and V. A. Tyumentsev, “Yank vector field and Yano-Killing tensor field in the flat and de-Sitter spaces,” Vestn. Omsk Univ., 3, 20–21 (2000).
Sh. Kobayashi, Transformation Groups in Differential Geometry, Springer-Verlag, Berlin-Heidelberg-New York (1972).
Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I, Interscience Publishers, New York-London-Sydney (1963).
Sh. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. II, Interscience Publishers, New York-London-Sydney (1969).
I. Kolai, P. W. Michor, and J. Slowak, Natural Operators in Differential Geometry, Springer-Verlag, Berlin-New York (1993).
M. Kora, “On conformal Killing forms and the proper space of for p-forms,” Math. J. Okayama Univ., 22, 195–204 (1980).
D. Kramer, H. Stephani, E. Herlt, and M. MacCallum, Exact Solutions of Einstein’s Field Equations, Cambridge Monogr. Math. Phys., 6, Cambridge Univ. Press (1980).
J. E. Marsden and F. J. Tipler, “Maximal hypersurfaces and foliations of constant mean curvature in general relativity,” Phys. Rep., 66, No. 3, 109–139 (1980).
R. Martens and D. P. Mason, “Kinematics and dynamic properties of conformal Killing vectors in anisotropic fluids,” J. Math. Phys., 27, No. 12, 2987–2994 (1986).
D. P. Mason and M. Tsamparlis, “Spacelike conformal Killing vector and spacelike congruences,” J. Math. Phys., 26, No. 11, 2881–2901 (1985).
D. Meyer, “Sur les variétés riemanniennes opérateur de coubure positif,” C. R. Acad. Sci. Paris, 272, 482–485 (1971).
C. W. Misner, K. S. Thorn, and J. A. Wheeler, Gravitation, W. H. Freeman, New York (1973).
S. Montiel, “An integral inequality for compact space-like hypersurfaces in de Sitter space and applications to the case of constant mean curvature,” Indiana Univ. Math. J., 37, No. 4, 909–917 (1988).
M. T. Mustafa, “Bochner technique for harmonic morphisms,” J. London Math. Soc., 57, No. 3, 746–756 (1998).
I. J. Muzinich, “Differential geometry in the large and compactification of higher-dimensional gravity,” J. Math. Phys., 27, No. 5, 1393–1397 (1986).
A. Nijenhuis, “A note on first integrals of geodesics,” Proc. Koninklijke Nederlandse Akademie van Wetenschappen, Ser. A, 70, No. 2, 141–145 (1967).
K. Nomizu, “What is affine differential geometry?” in: Proc. Conf. Differ. Geom., Münster (1982), pp. 42–43.
K. Nomizu, “On completeness in affine differential geometry,” Geom. Dedic., 20, No. 1, 43–49 (1986).
K. Nomizu, “A survey of recent result in affine differential geometry,” in: Geometry and Topology of Submanifolds (L. Verstraelen and A. West, Eds.), 3, World Scientific, London-Singapore (1991), pp. 227–256.
K. Nomizu, “On affine hypersurfaces with parallel nullity,” J. Math. Soc. Jpn., 44, No. 4, 693–699 (1992).
K. Nomizu and M. A. Magid, “On affine surfaces whose cubic forms are parallel relative to affine metric,” Proc. Jpn. Acad., Ser. A., 65, No. 7, 215–222 (1989).
K. Nomizu and U. Pinkall, “On the geometry of affine immersions,” Math. Z., 195, No. 2, 165–178 (1987).
K. Nomizu and T. Sasaki, Affine Differential Geometry, Cambridge Univ. Press, Cambridge (1994).
A. P. Norden, Affine Connection Spaces [in Russian], Nauka, Moscow (1976).
B. O’Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York-London (1983).
K. Ogiue and S. Tachibana, “Les varietés riemanniennes dont l’opérateur de courbure restreint est positif sont des sphéres d’holomogie réelle,” C. R. Acad. Sci. Paris, 289, 29–30 (1979).
H. K. Pak and T. Takahashi, “Harmonic forms in a compact contact manifold,” in: Proc. Fifth Pacific Geometry Conference, July 25–28, 2000, Tôhôku Univ. Press (2000), pp. 125–129.
R. S. Palais, Seminar on the Atiyah-Singer Index Theorem, Princeton Univ. Press, Princeton, New Jersey (1965).
P. Petersen, “Aspects of global Riemannian geometry,” Bull. Amer. Math. Soc., 36, No. 3, 297–344 (1999).
A. V. Pogorelov, “Complete affine-minimal hypersurfaces,” Dokl. Akad. Nauk SSSR, 301, No. 6, 1314–1316 (1988).
A. Polombo, “De nouvelles formules de Weitzenbock pour des endomorphismes harmoniques. Applications géométriques,” Ann. Sci Ec. Norm. Super., 25, No. 4, 393–428 (1992).
B. L. Reinhart, Differential Geometry of Foliations, Springer-Verlag, Berlin (1983).
G. de Rham, Differentiable Manifolds. Forms, Currents, Harmonic Forms, Springer-Verlag, Berlin (1984).
E. D. Rodionov and V. V. Slavskii, “Conformal and rank-one deformations of Riemannian metrics with area elements of zero curvature on compact manifolds,” in: Proc. Conf. “Geometry and Applications,” Novosibirsk, March 13–16, 2000, Novosibirsk (2000), pp. 171–182.
A. Romero and M. Sanchez, “An introduction to Bochner’s technique on Lorentzian manifolds,” in: Proc. V Fall Workshop: Differential Geometry and its Applications to Mathematical Physics, Jaca, Spain (1996), pp. 56–67.
A. Romero and M. Sanchez, “An integral inequality on compact Lorentz manifolds and its applications,” Bull. London Math. Soc., 28, 509–513 (1996).
A. Romero and M. Sanchez, “Bochner’s technique on Lorentz manifolds and infinitesimal conformal symmetries, Pac. J. Math., 186, No. 1, 141–148 (1998).
A. Romero and M. Sanchez, “Projective vector fields on Lorentzian manifolds,” Geom. Dedic., 93, 95–105 (2002).
L. A. Santal, “Affine integral geometry and convex bodies,” J. Microsc., 151, No. 3, 229–233 (1988).
J. A. Schouten, Ricci Calculus, Grundlehren Math. Wiss., 10. Springer-Verlag, Berlin (1954).
A. Schwenk, “Affinsphären mit ebenen Schttengrenzen,” in: Global Differential Geometry and Global Analysis 1984 (D. Ferus, R. B. Gardner, S. Helgason, and U. Simon, eds.), Lect. Notes Math., 1156, Springer-Verlag, Berlin (1985), pp. 296–315.
W. Seamon, “Harmonic 2-forms in four dimensions,” Proc. Amer. Math. Soc., 112, No. 2, 545–548 (1991).
Ya. L. Shapiro, “On one class of Riemannian spaces,” Tr. Semin. Vekt. Tenzor. Anal., 12, 203–212 (1963).
I. S. Shapiro and M. A. Ol’shanetskii, Lectures in Topology for Physicists [in Russian], Izhevsk (1999).
V. I. Shapovalov, “Symmetries of the Dirac-Fock equations,” Izv. Vyssh. Uchebn. Zaved., Ser. Fiz., 6, 57–63 (1975).
V. A. Sharafutdinov, Integral Geometry of Tensor Fields [in Russian], Nauka, Novosibirsk (1993).
R. N. Shcherbakov, Course of Affine and Projective Differential Geometry [in Russian], Tomsk University, Tomsk (1960).
B. Shiffman and A.-J. Sommese, Vanishing Theorems in Complex Manifolds, Progr. Math., 56, Birkhäuser, Boston (1985).
P. A. Shirokov, Selected Works in Geometry [in Russian], Kazan (1996), pp. 265–280.
P. A. Shirokov and A. P. Shirokov, Affine Differential Geometry [in Russian], Moscow (1959).
U. Simon, “Recent developments in affine differential geometry,” in: Proc. Int. Conf.: Differential Geometry and Its Applications, Dubrovnik, June 26–July 3, 1988, Inst. Math. Univ. Novi Sad (1989), pp. 327–347.
U. Simon, “Directly problems and the Laplacian in affine hypersurface theory,” Lect. Notes Math., 1369, 243–260, Springer-Verlag, Berlin (1989).
U. Simon and A. Schwenk, “Hypersurfaces with constant equiaffine mean curvature,” Arch. Math., 46, No. 1, 85–90 (1986).
U. Simon, A. Schwenk-Schellshmidt, and H. Viesel, Introduction to the Affine Differential Geometry of Hypersurfaces, Lect. Notes, Science Univ. Tokyo Press (1991).
J. Singh, General Relativity Theory [Russian translation], Inostr. Lit., Moscow (1963).
K. D. Singh, “Affine 2-Killing vector and tensor field,” Comp. Red. Acad. Bulg. Sci., 36, No. 11, 1375–1378 (1983).
E. N. Sinyukova, “On geodesic mappings of some special Riemannian spaces,” Mat. Zametki, 30, No. 6, 889–894 (1981).
W. Slysarska, “On devaluation from ample flatness,” Demonstr. Math., 21, No. 2, 505–511 (1988).
M. V. Smol’nikova, “On some property of Riemannian manifolds with sign-definite sectional curvature,” in: Modern problem of Field Theory [in Russian], Kazan (2000), pp. 365–367.
M. V. Smol’nikova, “On global geometry of harmonic symmetric bilinear differential forms,” in: Proc. Int. Conf. Differential Equations and Dynamical Systems, Vladimir, August 21–26, 2000, Vladimir (2000), pp. 87–88.
M. V. Smol’nikova, “Generalized recurrent symmetric tensor field,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 5, 48–51 (2002).
M. V. Smol’nikova, “On global geometry of harmonic symmetric bilinear forms,” Tr. Mat. Inst. Steklova, 202, 328–331 (2002).
M. V. Smol’nikova and S. E. Stepanov, “First-order fundamental differential operators on exterior and symmetric forms,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 11, 55–60 (2002).
M. V. Smol’nikova and S. E. Stepanov, “On a Yano differential operator,” in: Proc. Int. Conf. Differential Equations and Dynamical Systems, Vladimir (2002), pp. 129–131.
S. E. Stepanov, “Fields of symmetric tensors on compact Riemannian manifolds,” Mat. Zametki, 52, No. 4, 85–88 (1992).
S. E. Stepanov, “Bochner technique and cosmological models,” Izv. Vyssh. Uchebn. Zaved., Ser. Fiz., 6, 82–86 (1993).
S. E. Stepanov, “An integral formula for a Riemannian almost-product manifold,” Tensor, N.S., 55, 209–214 (1994).
S. E. Stepanov, “On an application of the theory of representations of groups in relativistic electrodynamics,” Izv. Vyssh. Uchebn. Zaved., Ser. Fiz., 5, 90–93 (1996).
S. E. Stepanov, “On the application of P. A. Shirokov’s theorem in the Bochner technique,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 9, 53–59 (1996).
S. E. Stepanov, “Killing forms on compact manifolds with boundary,” in: Proc. Int. Geom. Semin. “Modern Geometry and Theory of Physical Fields,” Kazan, Feb. 4–8, Kazan (1997), p. 114.
S. E. Stepanov, “A class of closed forms and special Maxwell’s equations,” Tensor, N.S., 58, 233–242 (1997).
S. E. Stepanov, “On the group-theoretic approach to the Einstein and Maxwell equations,” Teor. Mat. Fiz., 111, No. 1, 32–43 (1997).
S. E. Stepanov, “Bochner technique for m-dimensional compact manifolds with SL(m, ℝ)-structure,” Algebra Analiz, 10, No. 4, 703–714 (1998).
S. E. Stepanov, “Vector fields of conformal Killing forms on Riemannian manifolds,” Zap. Nauch. Semin. POMI, 261, 240–265 (1999).
S. E. Stepanov, “On isomorphisms of spaces of conformal Killing forms,” Diff. Geom. Mnogoobr. Figur, 31, 81–84 (2000).
S. E. Stepanov, “Bochner technique for physicists,” in: Lectures in Theoretical and Mathematical Physics [in Russian], Vol. 2, Kazan (2000), pp. 245–277.
S. E. Stepanov, “On some analytical method of general relativity,” Teor. Mat. Fiz., 122, No. 3, 482–496 (2000).
S. E. Stepanov, “New theorem of duality and its applications,” in: Modern Problems on Field Theory [in Russian], Kazan (2000), pp. 373–376.
S. E. Stepanov, “On conformal Killing 2-form of the electromagnetic field,” J. Geom. Phys., 33, 191–209 (2000).
S. E. Stepanov, “Riemannian almost product manifolds and submersions,” J. Math. Sci., 99, No. 6, 1788–1831 (2000).
S. E. Stepanov, “On some applications of the Stokes theorem in global Riemmanian geometry,” Fundam. Prikl. Mat., 8, No. 1, 1–18 (2002).
S. E. Stepanov, “On the Killing-Yano tensor,” Teor. Mat. Fiz., 134, No. 3, 380–385 (2003).
S. E. Stepanov, “New methods of the Bochner technique and their applications,” J. Math. Sci., 113, No. 3, 514–535 (2003).
S. E. Stepanov and I. G. Shandra, “Geometry of infinitesimal harmonic transformations,” Ann. Global Anal. Geom., 24, No. 3, 291–299 (2003).
S. E. Stepanov and I. I. Tsyganok, “Vector fields on Lorentz manifolds,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., 3, 81–83 (1994).
S. E. Stepanov and I. I. Tsyganok, “On a generalization of Kashiwada’s theorem,” in: Webs and Quasigroups, 1998–1999, Tver’ State Univ. (1999), pp. 162–167.
T. Sumitomo and K. Tandai, “Killing tensor fields on the standard sphere and spectra of SO(n + 1)/SO(n − 1) × SO(2) and O(n + 1)/O(n − 1) × O(2),” Osaka J. Math., 20, 51–78 (1983).
Sh. Tachibana, “On Killing tensor in a Riemannian space,” Tôhôku Math. J., 20, 257–264 (1968).
Sh. Tachibana, “On conformal Killing tensor in a Riemannian space,” Tôhôku Math. J., 21, 56–64 (1969).
Sh. Tachibana, “On projective Killing tensor,” Nat. Sci. Rep. Ochanomizu Univ., 21, 67–80 (1970).
K. Takano, “On projective Killing p-form in a Sasakian manifold,” Tensor, N.S., 60, 274–292 (1998).
K. Takano and S. Yamaguchi, “On a special projective Killing p-form with constant k in a Sasakian manifold,” Acta Sci. Math. (Szeged), 62, 299–317 (1996).
G. Thompson, “Killing tensor in spaces of constant curvature,” J. Math. Phys., 27, No. 11, 2693–2699 (1986).
V. V. Trofimov and A. T. Fomenko, “Riemannian geometry,” J. Math. Sci., 109, No. 2, 1345–1501 (2002).
G. Tsagas, “On the Killing tensor fields on a compact Riemannian manifold,” Balkan J. Geom. Appl., 1, No. 2, 91–97 (1996).
I. I. Tsyganok, “Torse-generating vector field and the affine homotety group,” in: Webs and Quasigroups [in Russian], Kalinin State University, Kalinin (1988), pp. 114–119.
I. I. Tsyganok, “Affine analogue of the Yano-Bochner method,” in: Proc. Rep. Conf., Sept. 21–22, 1990, Tartu Iniversity, Tartu (1990), pp. 76–78.
I. I. Tsyganok, Affine Geometry of Vector Fields [in Russian], Theses, MGPI, Moscow (1990).
I. I. Tsyganok, “Solenoidal vector fields on a compact manifold,” in: Proc. VI Int. Conf. of Women Mathematicians [in Russian], May 25–30, 1998, Cheboksary Univ., Cheboksary (1998), p. 68.
I. I. Tsyganok and S. E. Stepanov, “Bochner technique in affine differential geometry,” in: Algebraic Methods in Geometry [in Russian], Moscow (1992), pp. 50–55.
I. I. Tsyganok and S. E. Stepanov, “Vector fields in manifold with equiaffine connection,” in: Webs and Quasigroups, Tver State Univ. Press (1993), pp. 70–77.
I. I. Tsyganok and S. E. Stepanov, “The Hodge operator on a manifold with equiaffine structure,” Diff. Geom. Mnogoobr. Figur, 27, 114–117 (1996).
I. I. Tsyganok and S. E. Stepanov, “On a natural second-order differential operator,” in: Tr. Ross. Association “Women Mathematicians” [in Russian], 9, No. 1 (2001), pp. 68–71.
X.-J. Wang, “Affine maximal hypersurfaces,” in: Proc. Int. Congr. Math. Beijing, 2000, 3, Higher Educ. Press, Beijing (2000), pp. 221–231.
M. Weber, “Die Bochner-methode und sius Starrheitssatz,” Bonn. Math. Schr., 198, 1–58 (1989).
R. Weitzenbock, Invariantentheorie, Noordhoft, Groningen (1923).
J. A. Wolf, Spaces of Constant Curvature, McGraw-Hill, New York, etc. (1967).
N. M. J. Woodhouse, “Killing tensor and the separation of the Hamilton-Jacobi equation,” Commun. Math. Phys., 44, No. 9, pp. 1159–1167 (1975).
H. Wu, “The Bochner technique,” in: Proc. Beijinng Symp. Differential Geometry and Differential Equations, Aug. 18–Sept. 21, 1980, 2, Gordon and Breach, New York (1982), pp. 929–1071.
H. Wu, The Bochner technique in differential geometry, Math. Rep., 3, part 2, Hardwood Academic Publishers, London-Paris-New York (1988).
R. Xiaochun, “A Bochner theorem and applications,” Duke Math. J., 91, No. 2, 381–392 (1998).
L. Ximin, “Integral inequalities for maximal space-like submanifolds in the indefinite space form,” Bulkan J. Geom. Appl., 6, No. 1, 109–114 (2001).
K. Yano, Integral Formulas in Riemannian Geometry, Marcel Dekker, New York (1970).
K. Yano and S. Bochner, Curvature and Betti Numbers, Ann. Math. Stud., 32, Princeton Univ. Press, Princeton, New Jersey (1953).
Ch.-T. Yau and Ch.-Y. Cheng, “Complete affine hypersurfaces, I. The completeness of affine metrics,” Commun. Pure Appl. Math., 39, No. 6, 839–866 (1986).
G. Yun, “Total scalar curvature and L 2-harmonic 1-forms on minimal hypersurface in Euclidean space,” Geom. Dedic., 89, 135–141 (2002).
V. D. Zakharov, Gravitational Waves in the Einstein Gravitation Theory [in Russian], Nauka, Moscow (1972).
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Translated from Fundamental’naya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 11, No. 1, Geometry, 2005.
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Stepanov, S.E. Vanishing theorems in affine, Riemannian, and Lorentz geometries. J Math Sci 141, 929–964 (2007). https://doi.org/10.1007/s10958-007-0024-6
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DOI: https://doi.org/10.1007/s10958-007-0024-6