An elementary Treatise on Plane and Solid Geometry - download pdf or read online

By Benjamin Peirce

ISBN-10: 1425514650

ISBN-13: 9781425514655

It is a replica of a e-book released prior to 1923. This booklet could have occasional imperfections comparable to lacking or blurred pages, negative images, errant marks, and so on. that have been both a part of the unique artifact, or have been brought by way of the scanning approach. We think this paintings is culturally vital, and regardless of the imperfections, have elected to convey it again into print as a part of our carrying on with dedication to the upkeep of published works around the globe. We savor your realizing of the imperfections within the renovation approach, and desire you take pleasure in this priceless ebook.

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Example text

65 III. , denotes the Poincare M into the unit disk A={[G(~: lclO if p * q. For example if M = (c” then cM -= 0 by Liouville’s theorem. Similarly, cM = 0 on each compact complex manifold M, as by the maximum principle each holomorphic function on J4 is constant. On the other hand, cM apparently is a metric if M is a bounded domain in tl?. is easy to establish the contraction property of this semimetric with respect to holomorphic maps.

For example, for a spherical shell D = (z~d;“: 1 < 1z I< 2) with n > 1 by the theorem on removal of compact singularities (cf. [62]) cn coincides with the metric cs of the ball B= {lzl<2), so that the inner distance &(p, q) is larger than CD@, q) for suitable p, q. ViguC [71] has found a more subtle example: if D is the domain and one takes the points of holomorphy {z&‘: lz11+lz2~, 1z,z21<1/16) O=(O,O),z=(z,,z,),with 1/8

Together with each point of the domain also each circumference {zde: 0s 8s2 z} belongs to it). : C,(O, v)< l> coincides with the domain itself; b) every biholomorphic automorphism of the domain which preserves the origin is linear. The invariance of the Carathiodory metric is used in an essential manner in several results on the biholomo%hic nonequivalence of domains of different types. I. Pinchuk’s theorem [SS] on biholomorphic nonequivakmce of two bounded pseudoconvex domains in tI?. H. Khenkin’s theorem [37] on the biholomorphic nonequivalence of analytic polyhedra of a sufliciently general type with domains whose boundaries contain a nonempty open subset of points of strict pseudoconvexity.

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An elementary Treatise on Plane and Solid Geometry by Benjamin Peirce

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