| Differential Geometry |
Article Index for Differential |
Website Links For Differential Geometry |
Information AboutDifferential Geometry |
| CATEGORIES ABOUT DIFFERENTIAL GEOMETRY AND TOPOLOGY | |
| differential geometry | |
| differential topology | |
|
INTRINSIC VERSUS EXTRINSIC Initially and up to the middle of the s, Surface s were considered as lying in a Euclidean Space of higher dimension (for example a surface in an ambient space of three dimensions). The simplest results are those in the Differential Geometry Of Curves . Starting with the work of Riemann , the ''intrinsic'' point of view was developed, in which one cannot speak of moving 'outside' the geometric object because it is considered as given in a free-standing way. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it is harder to define the central concept of Curvature and other structures such as Connection s, so there is a price to pay. These two points of view can be reconciled, i.e. the extrinsic geometry can be considered as a structure additional to the intrinsic one (see the Nash Embedding Theorem ). TECHNICAL REQUIREMENTS The apparatus of differential geometry is that of ''calculus on manifolds'': this includes the study of . A Differential Manifold is a Topological Space with a collection of Homeomorphism s from open sets of the space to open subsets in Rn such that the open sets cover the space, and if '''f''', '''g''' are homeomorphisms then the function f o '''g''' -1 from an open subset of the open unit ball to the open unit ball is infinitely differentiable. We say a function from the manifold to R is infinitely differentiable if its composition with every homeomorphism results in an infinitely differentiable function from the open Unit Ball to R. At every point of the manifold, there is the Tangent Space at that point, which consists of every possible velocity (direction and magnitude) with which it is possible to travel away from this point. For an n-dimensional manifold, the tangent space at any point is an n-dimensional vector space, or in other words a copy of Rn. The tangent space has many definitions. One definition of the tangent space is as the dual space to the linear space of all functions which are zero at that point, divided by the space of functions which are zero and have a first derivative of zero at that point. Having a zero derivative can be defined by "composition by every differentiable function to the reals has a zero derivative", so it is defined just by differentiability. A Vector Field is a function from a manifold to the disjoint union of its tangent spaces (this union is itself a manifold known as the Tangent Bundle ), such that at each point, the value is an element of the tangent space at that point. Such a mapping is called a Section of a Bundle . A vector field is differentiable if for every differentiable function, applying the vector field to the function at each point yields a differentiable function. Vector fields can be thought of as time-independent differential equations. A differentiable function from the reals to the manifold is a curve on the manifold. This defines a function from the reals to the tangent spaces: the velocity of the curve at each point it passes through. A curve will be said to be a solution of the vector field if, at every point, the velocity of the curve is equal to the vector field at that point.
DIFFERENTIAL TOPOLOGY Differential topology per se considers the properties and structures that require only a smooth structure on a manifold to define (such as those in the previous section). Smooth manifolds are 'softer' than manifolds with extra geometric structures, which can act as obstructions to certain types of equivalences and deformations that exist in differential topology. For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold—that is, one can smoothly "flatten out" certain manifolds, but it might require distorting the space and affecting the curvature or volume. Conversely, smooth manifolds are more rigid than the topological manifolds. Certain topological manifolds have no smooth structures at all (see Donaldson's Theorem ) and others have more than one inequivalent smooth structure (such as Exotic Sphere s). Some constructions of smooth manifold theory, such as the existence of tangent bundles, can be done in the topological setting with much more work, and others cannot. BRANCHES OF DIFFERENTIAL GEOMETRY Contact geometry Contact Geometry is an analog of Symplectic Geometry which works for certain manifolds of odd dimension. Roughly, the contact structure on (2''n''+1)-dimensional manifold is a choice of a hyperplane field that is nowhere integrable. This is equivalent to the hyperplane field being defined by a 1-form such that does not vanish anywhere. Finsler geometry Finsler Geometry has the ''Finsler manifold '' as the main object of study — this is a differential manifold with a Finsler Metric , i.e. a Banach Norm defined on each tangent space. A Finsler metric is much more general structure than a Riemannian metric. Riemannian geometry Riemannian Geometry has Riemannian Manifold s as the main object of study — Smooth Manifold s with additional structure which makes them look ''infinitesimally'' like Euclidean Space . These allow one to generalise the notion from Euclidean geometry and analysis such as Gradient of a function, Divergence , Length of Curve s and so on; without assumptions that the space is ''globally'' so symmetric. The Riemannian Curvature Tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it is to being flat. Symplectic topology Symplectic topology is the study of ''symplectic manifolds'', which can occur only in even dimensions. A symplectic manifold is a differentiable manifold equipped with a symplectic form (that is, a Closed non-degenerate 2- Form ). Unlike in Riemannian geometry, all symplectic manifolds are locally isomorphic, so the only invariants of a symplectic manifold are global in nature. SEE ALSO
EXTERNAL LINKS
REFERENCE BOOKS #''A Comprehensive Introduction to Differential Geometry (5 Volumes), 3rd Edition'' by Michael Spivak (1999) #''Differential Geometry of Curves and Surfaces'' by Manfredo Do Carmo (1976). A classical geometric approach to differential geometry without the tensor machinery. #''Riemannian Geometry'' by Manfredo Perdigao do Carmo, Francis Flaherty (1994) #''Geometry from a Differentiable Viewpoint'' by John McCleary (1994) #''A First Course in Geometric Topology and Differential Geometry'' by Ethan D. Bloch (1996) #''Modern Differential Geometry of Curves and Surfaces with Mathematica'', 2nd ed. by Alfred Gray (1998) #''Applied Differential Geometry'' by William L. Burke (1985) |
|
|