Local and Global Properties of Hypersurfaces
Abstract
Differential Geometry is essentially the study of curves,
surfaces and hyper surfaces in
Rn.
Through the use of
analytic techniques, an understanding of the geometric
properties of these objects has led to profound breakthroughs
in the world of applied mathematics. The most basic form
which can be studied is known as a curve. These take the
form that their name might imply; they can be visualized
as lines that twist and turn through space with a dimension
less than two. Next is the object known as a surface.
Again, it can be visualized in everyday life as a coffee
cup, or the roof of a house, though they also can be much
more complicated. The final object of interest is the
hypersurface. It is analogous to our common understanding
of a regular surface in three dimensions. It is an object
in
Rn of degree
1 < d < n. In order to fully understand
these objects, both there local and global properties must
be considered. The local being that which occurs in the
neighborhood of a point, while the global considers
invariants over all of the domain.
Table of Contents
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