By Benjamin Peirce

ISBN-10: 1425514650

ISBN-13: 9781425514655

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**Extra resources for An elementary Treatise on Plane and Solid Geometry**

**Example text**

65 III. , denotes the Poincare M into the unit disk A={[G(~: lcl

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.

### An elementary Treatise on Plane and Solid Geometry by Benjamin Peirce

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