Volumes of certain small geodesic balls and almost-Hermitian geometry
Copyright policy )Volumes of certain small geodesic balls and almost-Hermitian geometry
Let (M,g) be a Riemannian manifold and let \(B_ m(r)\) be a small geodesic ball with center m and radius r. In [Acta Math. 142, 157-198 (1979; Zbl 0428.53017)] \textit{A. Gray} and the reviewer determined the first four nonzero coefficients in the power series expansion for the volume \(V_ m(r)\) of the ball \(B_ m(r)\). In [Compos. Math. 46, 121-132 (1982; Zbl 0489.53043)] the author determined the first two nonzero terms in the volume expansion for \(V_ m(r)\) when M is an arbitrary manifold equipped with an arbitrary metric connection D. In this paper the author considers the case of an almost Hermitian manifold and takes for D the characteristic connection. His purpose is to study the relation between the second term in the expansion and the sixteen classes of almost Hermitian manifolds determined by \textit{A. Gray} and \textit{L. Hervella} in [Ann. Mat. Pura Appl., IV. Ser. 123, 35-58 (1980; Zbl 0444.53032)]. Also, for some special compact complex manifolds, he studies the relation between this term and the spectrum of the complex Laplacian.
- University of Valencia Spain
power series expansion, almost Hermitian manifold, Spectral problems; spectral geometry; scattering theory on manifolds, General geometric structures on manifolds (almost complex, almost product structures, etc.), geodesic ball, Global differential geometry of Hermitian and Kählerian manifolds, complex manifolds, Laplacian, spectrum
power series expansion, almost Hermitian manifold, Spectral problems; spectral geometry; scattering theory on manifolds, General geometric structures on manifolds (almost complex, almost product structures, etc.), geodesic ball, Global differential geometry of Hermitian and Kählerian manifolds, complex manifolds, Laplacian, spectrum
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