In
mathematics, the Riemannian connection on a
surface
A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...
or
Riemannian 2-manifold refers to several intrinsic geometric structures discovered by
Tullio Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
,
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
and
Hermann Weyl in the early part of the twentieth century:
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
,
covariant derivative and
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
. These concepts were put in their current form with
principal bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equi ...
s only in the 1950s. The classical nineteenth century approach to the
differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
, due in large part to
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
, has been reworked in this modern framework, which provides the natural setting for the classical theory of the
moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay te ...
as well as the
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...
of higher-dimensional
Riemannian manifolds. This account is intended as an introduction to the theory of
connections.
Historical overview
After the classical work of Gauss on the
differential geometry of surfaces
In mathematics, the differential geometry of surfaces deals with the differential geometry of smooth surfaces with various additional structures, most often, a Riemannian metric.
Surfaces have been extensively studied from various perspective ...
and the subsequent emergence of the concept of
Riemannian manifold initiated by
Bernhard Riemann in the mid-nineteenth century, the geometric notion of
connection developed by
Tullio Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
,
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometr ...
and
Hermann Weyl in the early twentieth century represented a major advance in
differential geometry. The introduction of
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
,
covariant derivatives and
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
s gave a more conceptual and uniform way of understanding curvature, allowing generalisations to higher-dimensional manifolds; this is now the standard approach in graduate-level textbooks.
It also provided an important tool for defining new topological invariants called
characteristic classes via the
Chern–Weil homomorphism In mathematics, the Chern–Weil homomorphism is a basic construction in Chern–Weil theory that computes topological invariants of vector bundles and principal bundles on a smooth manifold ''M'' in terms of connections and curvature representing ...
.
Although Gauss was the first to study the differential geometry of surfaces in Euclidean space E
3, it was not until Riemann's Habilitationsschrift of 1854 that the notion of a Riemannian space was introduced. Christoffel introduced his eponymous symbols in 1869. Tensor calculus was developed by
Ricci Ricci () is an Italian surname, derived from the adjective "riccio", meaning curly. Notable Riccis Arts and entertainment
* Antonio Ricci (painter) (c.1565–c.1635), Spanish Baroque painter of Italian origin
* Christina Ricci (born 1980), Ameri ...
, who published a systematic treatment with
Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
in 1901. Covariant differentiation of tensors was given a geometric interpretation by who introduced the notion of parallel transport on surfaces. His discovery prompted
Weyl and
Cartan to introduce various notions of connection, including in particular that of affine connection. Cartan's approach was rephrased in the modern language of principal bundles by
Ehresmann, after which the subject rapidly took its current form following contributions by
Chern, Ambrose and
Singer
Singing is the act of creating musical sounds with the voice. A person who sings is called a singer, artist or vocalist (in jazz and/or popular music). Singers perform music (arias, recitatives, songs, etc.) that can be sung with or withou ...
,
Kobayashi
Kobayashi (written: lit. "small forest") is the 8th most common Japanese surname. A less common variant is . Notable people with the surname include:
Art figures
Film, television, theater and music
*, Japanese actress and voice actress
*, ...
,
Nomizu, Lichnerowicz and others.
Connections on a surface can be defined in a variety of ways. The Riemannian connection or
Levi-Civita connection is perhaps most easily understood in terms of lifting
vector fields, considered as first order
differential operators acting on functions on the manifold, to differential operators on sections of the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
. In the case of an embedded surface, this lift is very simply described in terms of orthogonal projection. Indeed, the vector bundles associated with the frame bundle are all sub-bundles of trivial bundles that extend to the ambient Euclidean space; a first order differential operator can always be applied to a section of a trivial bundle, in particular to a section of the original sub-bundle, although the resulting section might no longer be a section of the sub-bundle. This can be corrected by projecting orthogonally.
The Riemannian connection can also be characterized abstractly, independently of an embedding. The equations of geodesics are easy to write in terms of the Riemannian connection, which can be locally expressed in terms of the Christoffel symbols. Along a curve in the surface, the connection defines a
first order differential equation
In mathematics, an ordinary differential equation (ODE) is a differential equation whose unknown(s) consists of one (or more) function(s) of one variable and involves the derivatives of those functions. The term ''ordinary'' is used in contrast ...
in the frame bundle. The
monodromy of this equation defines
parallel transport
In geometry, parallel transport (or parallel translation) is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent b ...
for the connection, a notion introduced in this context by
Levi-Civita
Tullio Levi-Civita, (, ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus (tensor calculus) and its applications to the theory of relativity, but who also made signific ...
.
This gives an equivalent, more geometric way of describing the connection as lifting paths in the manifold to paths in the frame bundle. This formalises the classical theory of the "moving frame", favoured by French authors. Lifts of loops about a point give rise to the
holonomy group
In differential geometry, the holonomy of a connection on a smooth manifold is a general geometrical consequence of the curvature of the connection measuring the extent to which parallel transport around closed loops fails to preserve the geomet ...
at that point. The Gaussian curvature at a point can be recovered from parallel transport around increasingly small loops at the point. Equivalently curvature can be calculated directly infinitesimally in terms of
Lie brackets of lifted vector fields.
The approach of Cartan, using connection 1-forms on the
frame bundle
In mathematics, a frame bundle is a principal fiber bundle F(''E'') associated to any vector bundle ''E''. The fiber of F(''E'') over a point ''x'' is the set of all ordered bases, or ''frames'', for ''E'x''. The general linear group acts nat ...
of ''M'', gives a third way to understand the Riemannian connection, which is particularly easy to describe for an embedded surface. Thanks to a result of , later generalized by , the Riemannian connection on a surface embedded in Euclidean space ''E''
3 is just the pullback under the Gauss map of the Riemannian connection on ''S''
2.
Using the identification of ''S''
2 with the
homogeneous space SO(3)/SO(2), the connection 1-form is just a component of the
Maurer–Cartan 1-form on SO(3). In other words, everything reduces to understanding the 2-sphere properly.
Covariant derivative
For a surface ''M'' embedded in E
3 (or more generally a higher-dimensional Euclidean space), there are several equivalent definitions of a
vector field ''X'' on ''M'':
* a smooth map of ''M'' into E
3 taking values in the tangent space at each point;
* the
velocity vector
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
of a
local flow
In mathematics, a flow formalizes the idea of the motion of particles in a fluid. Flows are ubiquitous in science, including engineering and physics. The notion of flow is basic to the study of ordinary differential equations. Informally, a f ...
on ''M'';
* a first order
differential operator without constant term in any local chart on ''M'';
* a
derivation
Derivation may refer to:
Language
* Morphological derivation, a word-formation process
* Parse tree or concrete syntax tree, representing a string's syntax in formal grammars
Law
* Derivative work, in copyright law
* Derivation proceeding, a proc ...
of ''C''
∞(''M'').
The last condition means that the assignment ''f'' ''Xf'' on ''C''
∞(''M'') satisfies the
Leibniz rule
:
The space of all
vector fields
(''M'') forms a
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Mo ...
over ''C''
∞(''M''), closed under the
Lie bracket
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
:
with a ''C''
∞(''M'')-valued inner product (''X'',''Y''), which encodes the
Riemannian metric
In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space '' ...
on ''M''.
Since
(''M'') is a submodule of ''C''
∞(''M'', E
3)=''C''
∞(''M'')
E
3, the operator ''X''
''I'' is defined on
(''M''), taking values in ''C''
∞(''M'', E
3).
Let ''P'' be the smooth map from ''M'' into ''M''
3(R) such that ''P''(''p'') is the
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
of E
3 onto the tangent space at ''p''. Thus for the unit normal vector n
''p'' at ''p'', uniquely defined up to a sign, and v in E
3, the projection is given by (''p'')(v) = v - (v · n
''p'') n
''p''.
Pointwise multiplication by ''P'' gives a ''C''
∞(''M'')-module map of ''C''
∞(''M'', E
3) onto
(''M'') . The assignment
:
defines an operator
on
(''M'') called the covariant derivative, satisfying the following properties
#
is ''C''
∞(''M'')-linear in ''X''
#
(Leibniz rule for derivation of a module)
#
(
compatibility with the metric)
#
(symmetry property).
The first three properties state that
is an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
compatible with the metric, sometimes also called a ''hermitian'' or
metric connection. The last symmetry property says that the
torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a cur ...
: