Book Review
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MathSciNet review:
1567193
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Book Information:
Author:
Ethan Akin
Title:
The metric theory of Banach manifolds
Additional book information:
Lecture Notes in Math., vol. 662, Springer-Verlag, Berlin and New York, 1978, xix + 306 pp., $13.50.
M. Cantor, Sobolev inequalities for Riemannian bundles, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 171–184. MR 0380873
M. Cantor, Spaces of functions with asymptotic conditions on $R^{n}$, Indiana Univ. Math. J. 24 (1974/75), 897–902. MR 365621, DOI 10.1512/iumj.1975.24.24072
M. Cantor, Some problems of global analysis on asymptotically simple manifolds, Compositio Math. 38 (1979), no. 1, 3–35. MR 523260
J. Dowling, Finsler geometry on Sobolev manifolds, Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 1–10. MR 0263119
Halldór I. Elĭasson, Geometry of manifolds of maps, J. Differential Geometry 1 (1967), 169–194. MR 226681
Richard A. Graff, Elements of nonlinear functional analysis, Mem. Amer. Math. Soc. 16 (1978), no. 206, xii+196. MR 500408, DOI 10.1090/memo/0206
Richard S. Hamilton, Regularity theorems for partial differential operators, J. Differential Geometry 5 (1971), 39–58. MR 415670
Richard S. Hamilton, The inverse function theorem of Nash and Moser, Bull. Amer. Math. Soc. (N.S.) 7 (1982), no. 1, 65–222. MR 656198, DOI 10.1090/S0273-0979-1982-15004-2
David W. Henderson, Infinite-dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25–33. MR 250342, DOI 10.1016/0040-9383(70)90046-7
Thomas J. R. Hughes, Tosio Kato, and Jerrold E. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1976), no. 3, 273–294 (1977). MR 420024, DOI 10.1007/BF00251584
Tosio Kato, Quasi-linear equations of evolution, with applications to partial differential equations, Spectral theory and differential equations (Proc. Sympos., Dundee, 1974; dedicated to Konrad Jörgens), Lecture Notes in Math., Vol. 448, Springer, Berlin, 1975, pp. 25–70. MR 0407477
Nishan Krikorian, Differentiable structures on function spaces, Trans. Amer. Math. Soc. 171 (1972), 67–82. MR 312525, DOI 10.1090/S0002-9947-1972-0312525-5
J. Marsden, On product formulas for nonlinear semigroups, J. Functional Analysis 13 (1973), 51–72. MR 0355682, DOI 10.1016/0022-1236(73)90066-9
Richard S. Palais, Homotopy theory of infinite dimensional manifolds, Topology 5 (1966), 1–16. MR 189028, DOI 10.1016/0040-9383(66)90002-4
Richard S. Palais, Foundations of global non-linear analysis, W. A. Benjamin, Inc., New York-Amsterdam, 1968. MR 0248880
Richard S. Palais, Banach manifolds of fiber bundle sections, Actes du Congrès International des Mathématiciens (Nice, 1970) Gauthier-Villars, Paris, 1971, pp. 243–249. MR 0448405
Jean-Paul Penot, Topologie faible sur des variétés de Banach. Application aux géodésiques des variétés de Sobolev, J. Differential Geometry 9 (1974), 141–168 (French). MR 470987
18. P. Sobolevskiĭ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Trans. (2) 49 (1966), 1-62.
K. Uhlenbeck, Bounded sets and Finsler structures for manifolds of maps, J. Differential Geometry 7 (1972), 585–595. MR 334273
- 1.
- M. Cantor, Sobolev inequalities for Riemannian bundles, Proc. Sympos. Pure Math., vol. 27, Amer. Math. Soc., Providence, R.I., 1975, pp. 171-184. MR 0380873
- 1.
- M. Cantor, Spaces of functions with asymptotic conditions on Rn, Indiana Univ. Math. J. 24 (1975), 897-902. MR 365621
- 3.
- M. Cantor, Some problems of global analysis on asymptotically simple manifolds (preprint). MR 523260
- 4.
- J. Dowling, Finsler geometry on Sobolev manifolds, Proc. Sympos. Pure Math., vol. 15, Amer. Math. Soc., Providence, R.I., 1970, pp. 1-10. MR 263119
- 5.
- H. Eliasson, Geometry of manifolds of maps, J. Differential Geometry 1 (1967), 169-194. MR 226681
- 6.
- R. Graff, Elements of non-linear functional analysis, Mem. Amer. Math. Soc. No. 206, Amer. Math. Soc., Providence, R.I., 1978. MR 500408
- 7.
- R. Hamilton, Regularity theorems for partial differential operators, J. Differential Geometry 5 (1971), 39-58. MR 415670
- 8.
- R. Hamilton, The inverse function theorem of Nash and Moser (preprint). MR 656198
- 9.
- D. Henderson, Infinite dimensional manifolds are open subsets of Hilbert space, Topology 9 (1970), 25-33. MR 250342
- 10.
- T. Hughes, T. Kato and J. Marsden, Well-posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity, Arch. Rational Mech. Anal. 63 (1977), 273-294. MR 420024
- 11.
- T. Kato, Quasi-linear equations of evolution with applications to partial differential equations, Lecture Notes in Math., vol. 448, Springer-Verlag, Berlin and New York, 1975, pp. 25-70. MR 407477
- 12.
- N. Krikorian, Differentiable structures on function spaces, Trans. Amer. Math. Soc. 171 (1972), 67-82. MR 312525
- 13.
- J. Marsden, On product formulas for nonlinear semigroups, J. Functional Analysis 13 (1973), 51-72. MR 355682
- 14.
- R. Palais, Homotopy theory of infinite-dimensional manifolds, Topology 5 (1966) 1-16. MR 189028
- 15.
- R. Palais, Foundations of global non-linear analysis, Benjamin, New York, 1968. MR 248880
- 16.
- R. Palais, Banach manifolds of fiber bundle sections, Actes Congrés Internat Math. 2 (1970), 243-249. MR 448405
- 17.
- J. P. Penot, Topologie faible sur des variétés de Banach. Applications aux géodésiques des variétés de Sobolev, J. Differential Geometry 9 (1974), 141-168. MR 470987
- 18.
- P. Sobolevskiĭ, Equations of parabolic type in a Banach space, Amer. Math. Soc. Trans. (2) 49 (1966), 1-62.
- 19.
- K. Uhlenbeck, Bounded sets and Finsler structures for manifolds of maps, J. Differential Geometry 7 (1972), 585-595. MR 334273
Review Information:
Reviewer:
Richard Graff
Journal:
Bull. Amer. Math. Soc.
1 (1979), 960-965
DOI:
https://doi.org/10.1090/S0273-0979-1979-14700-1
