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Differential operators on Riemannian manifolds. (English) Zbl 0053.06502


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[1] See N. Aronszajn,Studies in Eigenvalue Problems, Report II, Office of Naval Research, N 9 onr 85101, 1950, and also Green’s functions and reproducing kernels,Proceedings of the Symposium on Spectral theory and Differential Problems; Oklahoma A. and M. College, 1951.
[2] For a complete discussion of differentiable manifolds see O. Veblen and J. H. C. Whitehead,Foundations of Differential Geometry, Cambridge, 1932. · JFM 58.0754.01
[3] For the theory of exterior differential forms see Elie Cartan,Les systèmes differentiels extérieurs et leurs applications géométriques, Paris, Hermann 1945. See also W. V. D. Hodge,Harmonic Integrals, Cambridge, 1941.
[4] For the theory of Riemannian manifolds we refer the reader to the classical textbooks of T. Levi-CivitaThe absolute differential calculus (translated from the Italian) Blackie and Son, Ltd. London and Glasgow, 1927 and L. P. Eisenhart,Riemannian Geometry Princeton Univ. Press, 1926.
[5] See Levi-Civita, —-loc cit pp. 153–154, (17’) and (17”).
[6] More details about such applications, especially to approximation methods, can be found in the following papers by N. Aronszajn: The Rayleigh-Ritz and A. Weinstein methods for approximation of eigenvalues. I and II,Proc. Nat. Acad. Sci. vol. 34 (1948) pp. 474–480, 594–601;Studies in eigenvalue problems, loc. cit. Reports II and III, Approximation methods for completely continuous symmetric operators,Proceedings of the Sympositium on Speciral Theory and Differential Problems, Oklahoma A. and M. College (1951), pp. 179–203. · Zbl 0038.24803
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