Differential geometry is a mathematical
discipline that uses the techniques of differential calculus
, integral calculus
, linear algebra
and multilinear algebra
to study problems in geometry
. The theory of plane and space curves
in the three-dimensional Euclidean space
formed the basis for development of differential geometry during the 18th century and the 19th century.
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifold
s. Differential geometry is closely related to differential topology
and the geometric aspects of the theory of differential equation
s. The differential geometry of surfaces
captures many of the key ideas and techniques endemic to this field.
History of development
Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in calculus
, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships.
When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms, the formal study of the nature of curves and surfaces became a field of study in its own right, with Monge
's paper in 1795, and especially, with Gauss
's publication of his article, titled 'Disquisitiones Generales Circa Superficies Curvas', in ''Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores'' in 1827.
Initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces.
Riemannian geometry studies Riemannian manifold
s, smooth manifold
s with a ''Riemannian metric''. This is a concept of distance expressed by means of a smooth 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, i.e. in the first order of approximation
. Various concepts based on length, such as the arc length
of plane regions, and volume
of solids all possess natural analogues in Riemannian geometry. The notion of a directional derivative
of a function from 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'', i.e. for small neighborhoods of points. Any two regular curves are locally isometric. However, the Theorema Egregium
of Carl Friedrich Gauss
showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvature
s at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor
is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat. An important class of Riemannian manifolds is the Riemannian symmetric space
s, whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean 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 has ''Finsler manifolds'' as the main object of study. This is a differential manifold with a ''Finsler metric'', that is, a Banach norm
defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. A Finsler structure on a manifold is a function such that:
# for all in and all ,
# is infinitely differentiable in ,
# The vertical Hessian of is positive definite.
is the study of symplectic manifold
s. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form
on each tangent space, i.e., a nondegenerate 2-form
''ω'', called the ''symplectic form''. A symplectic manifold is an almost symplectic manifold for which the symplectic form ''ω'' is closed: .
between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism
. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface
endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space
of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange
on analytical mechanics
and later in Carl Gustav Jacobi
's and William Rowan Hamilton
's formulations of classical mechanics
By contrast with Riemannian geometry, where the curvature
provides a local invariant of Riemannian manifolds, Darboux's theorem
states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. The first result in symplectic topology is probably the Poincaré–Birkhoff theorem
, conjectured by Henri Poincaré
and then proved by G.D. Birkhoff
in 1912. It claims that if an area preserving map of an annulus
twists each boundary component in opposite directions, then the map has at least two fixed points.
deals with certain manifolds of odd dimension. It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A ''contact structure'' on a -dimensional manifold ''M'' is given by a smooth hyperplane field ''H'' in the tangent bundle
that is as far as possible from being associated with the level sets of a differentiable function on ''M'' (the technical term is "completely nonintegrable tangent hyperplane distribution"). Near each point ''p'', a hyperplane distribution is determined by a nowhere vanishing 1-form
, which is unique up to multiplication by a nowhere vanishing function:
A local 1-form on ''M'' is a ''contact form'' if the restriction of its exterior derivative
to ''H'' is a non-degenerate two-form and thus induces a symplectic structure on ''H''''p''
at each point. If the distribution ''H'' can be defined by a global one-form
then this form is contact if and only if the top-dimensional form
is a volume form
on ''M'', i.e. does not vanish anywhere. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.
Complex and Kähler geometry
''Complex differential geometry'' is the study of complex manifolds
An almost complex manifold
is a ''real'' manifold
, endowed with a tensor
(1, 1), i.e. a vector bundle endomorphism
(called an ''almost complex structure
, such that
It follows from this definition that an almost complex manifold is even-dimensional.
An almost complex manifold is called ''complex'' if
is a tensor of type (2, 1) related to
, called the Nijenhuis tensor
(or sometimes the ''torsion'').
An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas
An ''almost Hermitian structure
'' is given by an almost complex structure ''J'', along with a Riemannian metric
''g'', satisfying the compatibility condition
An almost Hermitian structure defines naturally a differential two-form
The following two conditions are equivalent:
is the Levi-Civita connection
. In this case,
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 manifold
s) is given by all the smooth complex projective varieties
is the study of the intrinsic geometry of boundaries of domains in complex manifold
is the study of the set of angle-preserving (conformal) transformations on a space.
is the study of global geometric invariants without a metric or symplectic form.
Differential topology starts from the natural operations such as Lie derivative
of natural vector bundle
s and de Rham differential
. Beside Lie algebroid
s, also Courant algebroid
s start playing a more important role.
A Lie group
is a group
in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant vector field
s. Beside the structure theory there is also the wide field of representation theory
Gauge theory is the study of connections on vector bundles and principal bundles, and arises out of problems in mathematical physics
and physical gauge theories
which underpin the standard model of particle physics
. Gauge theory is concerned with the study of differential equations for connections on bundles, and the resulting geometric moduli space
s of solutions to these equations as well as the invariants that may be derived from them. These equations often arise as the Euler–Lagrange equations
describing the equations of motion of certain physical systems in quantum field theory
, and so their study is of considerable interest in physics.
Bundles and connections
The apparatus of vector bundle
s, principal bundle
s, and connection
s on bundles plays an extraordinarily important role in 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, i.e. a notion of parallel transport
. An important example is provided by affine connection
s. For a surface
, tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In Riemannian geometry
, the Levi-Civita connection
serves a similar purpose. (The Levi-Civita connection defines path-wise parallelism in terms of a given arbitrary Riemannian metric on a manifold.) More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be the space-time continuum
and the bundles and connections are related to various physical fields.
Intrinsic versus extrinsic
From the beginning and through the middle of the 19th century, differential geometry was studied from the ''extrinsic'' point of view: curve
s and 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
and differential geometry of surfaces
. 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 to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium
, to the effect that Gaussian curvature
is an intrinsic invariant.
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" of the universe?). However, there is a price to pay in technical complexity: the intrinsic definitions of curvature
become much less visually intuitive.
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
.) In the formalism of geometric calculus
both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator
Below are some examples of how differential geometry is applied to other fields of science and mathematics.
, differential geometry has many applications, including:
**Differential geometry is the language in which Albert Einstein
's general theory of relativity
is expressed. According to the theory, the universe is a smooth manifold equipped with a pseudo-Riemannian metric, which describes the curvature
. 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
are used in the study of electromagnetism
**Differential geometry has applications to both Lagrangian mechanics
and Hamiltonian mechanics
. Symplectic manifold
s in particular can be used to study Hamiltonian system
**Riemannian geometry and contact geometry have been used to construct the formalism of geometrothermodynamics
which has found applications in classical equilibrium thermodynamics
when modelling cell membrane structure under varying pressure.
, differential geometry has applications to the field of econometrics
(including computer graphics
) and computer-aided geometric design
draw on ideas from differential geometry.
, differential geometry can be applied to solve problems in digital signal processing
*In control theory
, differential geometry can be used to analyze nonlinear controllers, particularly geometric control
* In probability
, and information theory
, one can interpret various structures as Riemannian manifolds, which yields the field of information geometry
, particularly via the Fisher information metric
*In structural geology
, differential geometry is used to analyze and describe geologic structures.
*In computer vision
, differential geometry is used to analyze shapes.
*In image processing
, differential geometry is used to process and analyse data on non-flat surfaces.
's proof of the Poincaré conjecture
using the techniques of Ricci flow
s demonstrated the power of the differential-geometric approach to questions in topology
and it highlighted the important role played by its analytic methods.
* In wireless communications
, Grassmannian manifolds
are used for beamforming
techniques in multiple antenna
* Abstract differential geometry
* Affine differential geometry
* Analysis on fractals
* Basic introduction to the mathematics of curved spacetime
* Discrete differential geometry
* Glossary of differential geometry and topology
* Important publications in differential geometry
* Important publications in differential topology
* Integral geometry
* List of differential geometry topics
* Noncommutative geometry
* Projective differential geometry
* Synthetic differential geometry
* Systolic geometry
* Gauge theory (mathematics)
* B. Conrad. Differential Geometry handouts, Stanford UniversityA Modern Course on Curves and Surfaces, Richard S Palais, 2003Richard Palais's 3DXM Surfaces GalleryN. J. Hicks, Notes on Differential Geometry, Van Nostrand.MIT OpenCourseWare: Differential Geometry, Fall 2008