Doctoral thesis, Durham University. We examine some generic features of surfaces in the Euclidean 3-space related to the Gauss map on the surface.
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We consider these features on smooth surfaces and on singular surfaces with a cross-cap singularity. We study some symmetries between two classical pairs of foliations defined on smooth surfaces in : the asymptotic curves and the characteristic curves called harmonic mean curvature lines in. The asymptotic curves exist in hyperbolic regions of surfaces and have been well studied.
The characteristic curves are in certain ways the analogy of the asymptotic curves in elliptic regions. In this thesis we extend this analogy. We use We produce results on the characteristic curves mirroring those of Uribe-Vargas on the asymptotic curves. By considering cross-ratios of Legendrian lines in the manifold of contact elements to the surface we show that certain properties of the characteristic curves are invariant under projective transformations, and examine their behaviour at cusps of Gauss.
We establish an analogy of the Beltrami-Enepper Theorem, which allows us to distinguish between the two characteristic foliations in a natural geometric way. We show that the local properties of characteristic curves may be used to prove certain global results concerning the elliptic regions of smooth surfaces. Motivated by the study of the asymptotic, principal and characteristic curves on surfaces in , we construct a natural one-to-one correspondence between the set of non-degenerate binary differential equations BDEs and linear involutions on the real projective line.
We show that one may construct pairs of BDEs that have various symmetric properties using a single involution on. We study the folded singularities of BDEs, and associate an affine invariant to such points. Share Give access Share full text access. Share full text access. Please review our Terms and Conditions of Use and check box below to share full-text version of article. Abstract This paper gives a topological characterization of Diophantine and recurrent laminations on surfaces.
Volume 6 , Issue 2 June Pages Related Information. Close Figure Viewer. Browse All Figures Return to Figure. Previous Figure Next Figure. Pure Math. Mumford, J. Fogarty and F. Kirwan, Geometric invariant theory, 3rd ed.
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Pereira, On the height of foliated surfaces with vanishing Kodaira dimension, Publ. Serrano, Isotrivial fibred surfaces, Ann. Pura Appl. Siu, Invariance of plurigenera, Invent.
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Pairs of geometric foliations of regular and singular surfaces
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