Differential geometry Totally Explained

Everything about Differential Geometry totally explained

Differential geometry is a mathematical discipline that uses the methods of differential and integral calculus to study problems in geometry. The theory of plane and space curves and of surfaces in the three-dimensional Euclidean space formed the basis for its initial development in the eighteenth and nineteenth century. Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. It is closely related with differential topology and with the geometric aspects of the theory of differential equations. The proof of the Poincare conjecture using the techniques of Ricci flow demonstrated the power of the differential-geometric approach to questions in topology and highlighted the important role played by the analytic methods.

Branches of differential geometry

Riemannian geometry

Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric, a notion of a distance expressed by means of a positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point "infinitesimally", for example in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all admit natural analogues in Riemannian geometry. The notion of a directional derivative of a function from the multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor. Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally, for example for small neighborhoods of points. Any two regular curves are locally isometric. However, Theorema Egregium of Gauss showed that already for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated to a Riemannian manifold that measures how close it's to being flat. An important class of Riemannian manifolds is formed by the Riemannian symmetric spaces, whose curvature is constant. They are the closest to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry.

Pseudo-Riemannian geometry

Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold which is the mathematical basis of Einstein's general relativity theory of gravity.

Finsler geometry

Finsler geometry has the Finsler manifold as the main object of study — this is a differential manifold with a Finsler metric, for example a Banach norm defined on each tangent space. A Finsler metric is a much more general structure than a Riemannian metric. A Finsler structure on a manifold M is a function F : TM → [0,∞) such that:

F(x, my) = mF(x,y) for all x, y in TM,
F is infinitely differentiable in TM − domega=0,
$abla J=0,$

where

abla

is the Levi-Civita connection of

g

. In this case,

(J, g)

is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure. In particular, a Kähler manifold is both a complex and a symplectic manifold. A large class of Kähler manifolds (the class of Hodge manifolds) is given by all the smooth complex projective varieties.

CR geometry

CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds.

Bundles and connections

The apparatus of vector bundles, principal bundles, and connections on them plays an extraordinarily important role in the modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires in addition some way to relate the tangent spaces at different points, for example a notion of parallel transport. An important example is provided by affine connections. For a surface in R³, tangent planes at different points can be identified using the flat nature of the ambient Euclidean space. In Riemannian geometry, the Levi-Civita connection serves a similar purpose. More generally, differential geometers consider spaces with a vector bundle and a connection as a replacement for the notion of a Riemannian manifold. In this approach, the bundle is external to the manifold and has to be specified as a part of the structure, while the connection provides a further enhancement. In physics, the manifold may be the spacetime and bundles and connections correspond to various physical fields.

Intrinsic versus extrinsic

Initially and up to the middle of the nineteenth century, differential geometry was studied from the extrinsic point of view: curves and surfaces 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 can't speak of moving 'outside' the geometric object because it's considered as given in a free-standing way.
The intrinsic point of view is more flexible. For example, it's useful in relativity where space-time can't naturally be taken as extrinsic (what would be 'outside' it?). With the intrinsic point of view it's harder to define the central concept of curvature and other structures such as connections, so there's a price to pay.
These two points of view can be reconciled, for example the extrinsic geometry can be considered as a structure additional to the intrinsic one. (See the Nash embedding theorem.)

Applications of differential geometry

Below are some examples of how differential geometry is applied to other fields of science and mathematics.

In physics, differential geometry is the language in which Einstein's general theory of relativity is expressed. According to the theory, the universe is a smooth manifold equipped with pseudo-Riemannian metric, which described the curvature of space-time. Understanding this curvature is essential for the positioning of satellites into orbit around the earth. Differential geometry is also indispensable in the study of gravitational lensing and black holes.

In economics, differential geometry has applications to the field of econometrics.

Geometric modeling (including computer graphics) and computer-aided geometric design draw on ideas from differential geometry.

In engineering, differential geometry can be applied to solve problems in digital signal processing .

In physics, the use of differential forms is useful in the study of electromagnetism.

In physics, differential geometry has applications to both Lagrangian mechanics and Hamiltonian mechanics. Symplectic manifolds in particular can be used to study Hamiltonian systems.Further Information

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