Download Curves and Manifolds

Curves and Manifolds
In mathematics, a plane curve is a curve in a Euclidian plane (cf. space curve). The most
frequently studied cases are smooth plane curves (including piecewise smooth plane
curves), and algebraic plane curves.
A smooth plane curve is a curve in a real Euclidian plane R2 is a one-dimensional smooth
manifold. Equivalently, a smooth plane curve can be given locally by an equation f(x,y) =
0, where f is a smooth function of two variables, and the partial derivatives fx and fy are
not simultaneously equal to 0. In other words, a smooth plane curve is a plane curve
which "locally looks like a line" with respect to a smooth change of coordinates.
An algebraic plane curve is a curve in an affine or projective plane given by one
polynomial equation f(x,y) = 0 (or f(x,y,z) = 0, where f is a homogeneous polynomial, in
the projective case.)
Algebraic curves were studied extensively in the 18th to 20th centuries, leading to a very
rich and deep theory. The founders of the theory are Issac Newton, Bernhard Riemann
et.al., with some main contributors being Niels Henrik Abel, Henri Poincaré, Max
Noether, et.al. Every algebraic plane curve has a degree, which can be defined, in case of
an algebraically closed field, as number of intersections of the curve with a generic line.
For example, a circle x2 + y2 = 1 has degree 2.
An important classical result states that every non-singular plane curve of degree 2 in a
projective plane is isomorphic to the projection of the circle x2 + y2 = 1. However, the
theory of plane curves of degree 3 is already very deep, and connected with the
Weierstrass's theory of bi-periodic complex analytic functions (cf. elliptic curves,
Weierstrass P-function).
There are many questions in the theory of plane algebraic curves for which the answer is
not known as of the beginning of the 21st century.
A differentiable manifold is a type of manifold that is locally similar enough to Euclidean
space to allow one to do calculus. Any manifold can be described by a collection of
charts, also known as an atlas. One may then apply ideas from calculus while working
within the individual charts, since these lie in Euclidean spaces to which the usual rules
of calculus apply. If the charts are suitably compatible (namely, the transition from one
chart to another is differentiable), then computations done in one chart are valid in any
other differentiable chart.
More formally, a differentiable manifold is a topological manifold with a globally
defined differential structure. Any topological manifold can be given a differentiable
structure locally by using the homeomorphisms in its atlas and the standard differential
structure on Euclidean space. To induce a global differentiable structure on the local
coordinate systems induced by the homeomorphisms their composition on chart
intersections in the atlas must be differentiable functions on Euclidean space. In other
words, where the domains of charts overlap, the coordinates defined by each chart are
required to be differentiable with respect to the coordinates defined by every chart in the
atlas. The maps that relate the coordinates defined by the various charts to one another
are called transition maps.
The ability to induce such a differential structure allows one to extend the definition of
differentiability to spaces without global coordinate systems. Specifically, a differentiable
structure allows one to define globally differentiable tangent space, differentiable
functions, and differentiable tensor and vector fields. Differentiable manifolds are very
important in physics. Special kinds of differentiable manifolds form the basis for physical
theories such as classical mechanics, general relativity, and Yang-Mills theory. It is
possible to develop a calculus for differentiable manifolds. This leads to such
mathematical machinery as the exterior calculus. The study of calculus on differentiable
manifolds is known as differential geometry.
A topological manifold is a second countable Hausdorff space which is locally
homeomorphic to Euclidean space, by a collection (called an atlas) of homeomorphisms
called charts. The composition of one chart with the inverse of another chart is a function
called a transition map, and defines a homeomorphism of an open subset of Euclidean
space onto another open subset of Euclidean space.
There are a number of different types of differentiable manifolds, depending on the
precise differentiability requirements on the transition functions. Some common
examples include the following.

A differentiable manifold is a topological manifold equipped with an atlas whose
transition maps are all differentiable. More generally a Ck-manifold is a
topological manifold with an atlas whose transition maps are all k-times
continuously differentiable.

A smooth manifold or C∞-manifold is a differentiable manifold for which all the
transitions maps are smooth. That is derivatives of all orders exist; so it is a Ckmanifold for all k.

An analytic manifold, or Cω-manifold is a smooth manifold with the additional
condition that each transition map is analytic: the Taylor expansion is absolutely
convergent on some open ball.

A complex manifold is a topological space modeled on a Euclidean space over the
complex field and for which all the transition maps are holomorphic.
These definitions, however, leave out an important feature. They each still involve a
preferred choice of atlas. Given a differentiable manifold (in any of the above senses),
there is a notion of when two atlases are equivalent. Then, strictly speaking, a
differentiable manifold is an equivalence class of such atlases.