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Zeta Functions on the Unitary Sphere

  • S. Minakshisundaram (a1)

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In an earlier paper [5], the author defined a zeta function on the real sphere , whereas in the present paper it is proposed to define one on the unitary sphere where x i's are complex numbers and their complex conjugates. Following E. Cartan, harmonics on the unitary sphere are defined and then a zeta function formed just as in the case of a real sphere. The unitary sphere is seen to behave like an even-dimensional closed manifold, since results similar to the ones proved by the author and A. Pleijel [6] for closed manifolds (of even dimensions) are observed here also.

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References

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1. Bailey, W. N., Generalized hypergeometric series, Cambridge Tract No. 32 (1935).
2. Cartan, E., Leçons sur la géométrie projective complexe (Paris, 1931).
3. Cartan, E., Sur la détermination d'un systéme orthogonal complet dans un espace de Riemann symétrique clos, Rend. Cire. Mat. di Palermo, vol. 53 (1929), 217252.
4. Casimir, H. B. G., Rotation of a rigid body in quantum mechanics, Leiden thesis (1931).
5. Minakshisundaram, S., Zeta function on the sphere, J. Ind. Math. Soc, vol. 13 (1949), 4148.
6. Minakshisundaram, S. and Pleijel, A., Some properties of the eigenfunction of the Laplace operator on Riemann manifolds, Can. J. Math., vol. 1 (1949), 242286.
7. Weyl, H., Harmonics on homogeneous manifolds, Annals of Math., vol. 35 (1934), 485499.
8. Weyl, H., Ramification, old and new, of the eigenvalue problem, Bull. Amer. Math. Soc, vol. 56 (1950), 15139.
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