Volumes of tubes about complex submanifolds of complex projective space
HTML articles powered by AMS MathViewer
- by Alfred Gray PDF
- Trans. Amer. Math. Soc. 291 (1985), 437-449 Request permission
Abstract:
Simple formulas in terms of Chern classes are given for the volume of a tube about a Kähler submanifold of a space of constant holomorphic sectional curvature. A comparison theorem which generalizes these formulas is also given. Tubes about complete intersections in complex projective space are studied in detail.References
- Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
- Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
- F. J. Flaherty, The volume of a tube in complex projective space, Illinois J. Math. 16 (1972), 627–638. MR 315638
- Alfred Gray, A generalization of F. Schur’s theorem, J. Math. Soc. Japan 21 (1969), 454–457. MR 248681, DOI 10.2969/jmsj/02130454
- Alfred Gray, Chern numbers and curvature, Amer. J. Math. 100 (1978), no. 3, 463–476. MR 493905, DOI 10.2307/2373833
- Alfred Gray, Comparison theorems for the volumes of tubes as generalizations of the Weyl tube formula, Topology 21 (1982), no. 2, 201–228. MR 642000, DOI 10.1016/0040-9383(82)90005-2 —, Volumes of tubes about Kähler submanifolds as expressed in terms of Chern classes, J. Math. Soc. Japan 36 (1983).
- Alfred Gray, A product formula for the volumes of tubes about Kähler submanifolds of complex Euclidean space, Riv. Mat. Univ. Parma (4) 10 (1984), no. special vol. 10*, 109–121. MR 777318
- A. Gray and L. Vanhecke, The volumes of tubes in a Riemannian manifold, Rend. Sem. Mat. Univ. Politec. Torino 39 (1981), no. 3, 1–50 (1983). MR 706043
- Phillip A. Griffiths, Complex differential and integral geometry and curvature integrals associated to singularities of complex analytic varieties, Duke Math. J. 45 (1978), no. 3, 427–512. MR 507455
- Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Pure and Applied Mathematics, Wiley-Interscience [John Wiley & Sons], New York, 1978. MR 507725
- F. Hirzebruch, Topological methods in algebraic geometry, Third enlarged edition, Die Grundlehren der mathematischen Wissenschaften, Band 131, Springer-Verlag New York, Inc., New York, 1966. New appendix and translation from the second German edition by R. L. E. Schwarzenberger, with an additional section by A. Borel. MR 0202713 W. Katz, Die Chern-Formen von kählerschen Hyperflächen komplexer Raumformen, Inaugural Dissertation, Köln, 1979.
- John W. Milnor and James D. Stasheff, Characteristic classes, Annals of Mathematics Studies, No. 76, Princeton University Press, Princeton, N. J.; University of Tokyo Press, Tokyo, 1974. MR 0440554
- David Mumford, Algebraic geometry. I, Grundlehren der Mathematischen Wissenschaften, No. 221, Springer-Verlag, Berlin-New York, 1976. Complex projective varieties. MR 0453732
- E. Study, Kürzeste Wege im komplexen Gebiet, Math. Ann. 60 (1905), no. 3, 321–378 (German). MR 1511309, DOI 10.1007/BF01457616
- Hermann Weyl, On the Volume of Tubes, Amer. J. Math. 61 (1939), no. 2, 461–472. MR 1507388, DOI 10.2307/2371513
- W. Wirtinger, Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde in euklidischer und Hermitescher Maßbestimmung, Monatsh. Math. Phys. 44 (1936), no. 1, 343–365 (German). MR 1550581, DOI 10.1007/BF01699328
- Robert A. Wolf, The volume of tubes in complex projective space, Trans. Amer. Math. Soc. 157 (1971), 347–371. MR 281237, DOI 10.1090/S0002-9947-1971-0281237-8
Additional Information
- © Copyright 1985 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 291 (1985), 437-449
- MSC: Primary 53C65; Secondary 53C40
- DOI: https://doi.org/10.1090/S0002-9947-1985-0800247-2
- MathSciNet review: 800247