Athwartness
Athwartness
This thesis is a study of a number of problems in differential geometry and topology. Most of these concern smooth immersions of n-dimensionalImanifolds in (n+l)-dimensional Euclidean space. The first main theorem gives a simple algebraic formula which establishes a relation between the number of points in which.an immersion f of a smooth surface M in R3 intersects a given straight line L and the number of points m for which the affine tangent plane to f(M) at f(m) contains L. This theorem may be interpreted as a development of the theory of horizon immersions for surfaces in R3.An essential ingredient of many arguments is the concept of radial map of an immersion with respect to a given centre in Rn+l. The meaning of the critical set of such radial maps is explained in terms of the geometry of the immersion.The notion of athwartness for pairs of immersions in Rn+1 dominates the later parts of the thesis. Athwartness generalises the notion of horizon immersion, and only a few cases are, at present, tractable. These include the case of a pair of immersed hypersurfaces, and the case of an immersed curve and an immersed hypersurface. In both cases, necessary conditions for athwartness are obtained.Limited progress on such problems can be made using the methods of classical algebraic topology. An account of typical results of this kind is given in some detail.
University of Southampton
Craveiro de Carvalho, Francisco Jose
1978
Craveiro de Carvalho, Francisco Jose
Craveiro de Carvalho, Francisco Jose
(1978)
Athwartness.
University of Southampton, Doctoral Thesis.
Record type:
Thesis
(Doctoral)
Abstract
This thesis is a study of a number of problems in differential geometry and topology. Most of these concern smooth immersions of n-dimensionalImanifolds in (n+l)-dimensional Euclidean space. The first main theorem gives a simple algebraic formula which establishes a relation between the number of points in which.an immersion f of a smooth surface M in R3 intersects a given straight line L and the number of points m for which the affine tangent plane to f(M) at f(m) contains L. This theorem may be interpreted as a development of the theory of horizon immersions for surfaces in R3.An essential ingredient of many arguments is the concept of radial map of an immersion with respect to a given centre in Rn+l. The meaning of the critical set of such radial maps is explained in terms of the geometry of the immersion.The notion of athwartness for pairs of immersions in Rn+1 dominates the later parts of the thesis. Athwartness generalises the notion of horizon immersion, and only a few cases are, at present, tractable. These include the case of a pair of immersed hypersurfaces, and the case of an immersed curve and an immersed hypersurface. In both cases, necessary conditions for athwartness are obtained.Limited progress on such problems can be made using the methods of classical algebraic topology. An account of typical results of this kind is given in some detail.
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Published date: 1978
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Local EPrints ID: 459968
URI: http://eprints.soton.ac.uk/id/eprint/459968
PURE UUID: 25874d5f-b9f8-4d11-ade2-53b854aee597
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Date deposited: 04 Jul 2022 17:30
Last modified: 04 Jul 2022 17:30
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Author:
Francisco Jose Craveiro de Carvalho
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