Levi-Civita connection
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In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of
general relativity General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics ...
), the Levi-Civita connection is the unique
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 ...
on the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of a
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
(i.e.
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 ...
) that
preserves Fruit preserves are preparations of fruits whose main preserving agent is sugar and sometimes acid, often stored in glass jars and used as a condiment or spread. There are many varieties of fruit preserves globally, distinguished by the meth ...
the (
pseudo- The prefix pseudo- (from Greek ψευδής, ''pseudes'', "false") is used to mark something that superficially appears to be (or behaves like) one thing, but is something else. Subject to context, ''pseudo'' may connote coincidence, imitation, ...
)
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 ''T ...
and is
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
-free. The
fundamental theorem of Riemannian geometry In the mathematical field of Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique affine connection that is torsion-free and metric-compatible ...
states that there is a unique connection which satisfies these properties. In the theory of Riemannian and
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
s the term covariant derivative is often used for the Levi-Civita connection. The components (structure coefficients) of this connection with respect to a system of local coordinates are called
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
.


History

The Levi-Civita connection is named after
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 significa ...
, although originally "discovered" by
Elwin Bruno Christoffel Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provi ...
. Levi-Civita, along with
Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the ...
, used Christoffel's symbols to define the notion 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 (vector bundle), c ...
and explore the relationship of parallel transport with the
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonic ...
, thus developing the modern notion of
holonomy 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 ...
. In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. In 1906,
L. E. J. Brouwer Luitzen Egbertus Jan Brouwer (; ; 27 February 1881 – 2 December 1966), usually cited as L. E. J. Brouwer but known to his friends as Bertus, was a Dutch mathematician and philosopher, who worked in topology, set theory, measure theory and compl ...
was the first
mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...
to consider the
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 (vector bundle), c ...
of a
vector Vector most often refers to: *Euclidean vector, a quantity with a magnitude and a direction *Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism Vector may also refer to: Mathematic ...
for the case of a space of
constant curvature In mathematics, constant curvature is a concept from differential geometry. Here, curvature refers to the sectional curvature of a space (more precisely a manifold) and is a single number determining its local geometry. The sectional curvature i ...
. In 1917,
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 ...
pointed out its importance for the case of a
hypersurface In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean ...
immersed in a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, i.e., for the case of a
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real manifold, real, smooth manifold ''M'' equipped with a positive-definite Inner product space, inner product ...
embedded in a "larger" ambient space. He interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space. The Levi-Civita notions of
intrinsic derivative In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a different ...
and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding M^n \subset \mathbf^. In 1918, independently of Levi-Civita,
Jan Arnoldus Schouten Jan Arnoldus Schouten (28 August 1883 – 20 January 1971) was a Dutch mathematician and Professor at the Delft University of Technology. He was an important contributor to the development of tensor calculus and Ricci calculus, and was one of the ...
obtained analogous results. In the same year,
Hermann Weyl Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
generalized Levi-Civita's results.


Notation

* denotes a Riemannian or
pseudo-Riemannian manifold In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the ...
. * is the
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
of . * is the Riemannian or
pseudo-Riemannian metric In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
of . * are smooth vector fields on , i. e. smooth
section Section, Sectioning or Sectioned may refer to: Arts, entertainment and media * Section (music), a complete, but not independent, musical idea * Section (typography), a subdivision, especially of a chapter, in books and documents ** Section sign ...
s of . * is 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 ...
of and . It is again a smooth vector field. The metric can take up to two vectors or vector fields as arguments. In the former case the output is a number, the (pseudo-)
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
of and . In the latter case, the inner product of is taken at all points on the manifold so that defines a smooth function on . Vector fields act (by definition) as differential operators on smooth functions. In local coordinates (x_1,\ldots, x_n) , the action reads :X(f) = X^i\fracf = X^i\partial_i f where Einstein's
summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
is used.


Formal definition

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 ...
is called a Levi-Civita connection if # ''it preserves the metric'', i.e., . # ''it is
torsion Torsion may refer to: Science * Torsion (mechanics), the twisting of an object due to an applied torque * Torsion of spacetime, the field used in Einstein–Cartan theory and ** Alternatives to general relativity * Torsion angle, in chemistry Bi ...
-free'', i.e., for any vector fields and we have , where is 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 ...
of the vector fields and . Condition 1 above is sometimes referred to as compatibility with the metric, and condition 2 is sometimes called symmetry, cf. Do Carmo's text.


Fundamental theorem of (pseudo) Riemannian Geometry

Theorem Every pseudo Riemannian manifold (M,g) has a unique Levi Civita connection \nabla. ''proof'': If a Levi-Civita connection exists, it must be unique. To see this, unravel the definition of the action of a connection on tensors to find : X\bigl(g(Y,Z)\bigr) = (\nabla_X g)(Y, Z) + g(\nabla_X Y, Z) + g( Y, \nabla_X Z). Hence we can write condition 1 as : X\bigl(g(Y,Z)\bigr) = g(\nabla_X Y, Z) + g( Y, \nabla_X Z). By the symmetry of the metric tensor g we then find: : X \bigl(g(Y,Z)\bigr) + Y \bigl(g(Z,X)\bigr) - Z \bigl(g(Y,X)\bigr) = g(\nabla_X Y + \nabla_Y X, Z) + g(\nabla_X Z - \nabla_Z X, Y) + g(\nabla_Y Z - \nabla_Z Y, X). By condition 2, the right hand side is therefore equal to : 2g(\nabla_X Y, Z) - g( ,Y Z) + g( ,Z Y) + g( ,Z X), and we find the Koszul formula : g(\nabla_X Y, Z) = \tfrac \Big\. Hence, if a Levi-Civita connection exists, it must be unique, because Z is arbitrary, g is non degenerate, and the right hand side does not depend on \nabla. To prove existence, note that for given vector field X and Y, the right hand side of the Koszul expression is function-linear in the vector field Z, not just real linear. Hence by the non degeneracy of g, the right hand side uniquely defines some new vector field which we suggestively denote \nabla_X Y as in the left hand side. By substituting the Koszul formula, one now checks that for all vector fields X, Y,Z, and all functions f : g(\nabla_X (Y_1 + Y_2), Z) = g(\nabla_X Y_1, Z) + g(\nabla_X Y_2 , Z) : g(\nabla_X (f Y), Z) = X(f) g(Y, Z) + f g(\nabla_X Y,Z) : g(\nabla_X Y, Z) + g(\nabla_X Z, Y) = X\bigl(g(Y,Z)\bigr) : g(\nabla_X Y, Z) - g(\nabla_Y X, Z) = g( ,Y Z). Hence the Koszul expression does, in fact, define a connection, and this connection is compatible with the metric and is torsion free, i.e. is a (hence the) Levi-Civita connection. Note that with minor variations the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.


Christoffel symbols

Let \nabla be an affine connection on the tangent bundle. Choose local coordinates x^1, \ldots, x^n with coordinate basis vector fields \partial_1, \ldots, \partial_n and write \nabla_j for \nabla_. The
Christoffel symbols In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distanc ...
\Gamma^l_ of \nabla with respect to these coordinates are defined as : \nabla_j\partial_k = \Gamma^l_ \partial_l The Christoffel symbols conversely define the connection \nabla on the coordinate neighbourhood because : \begin \nabla_X Y &= \nabla_ (Y^k \partial_k) \\&= X^j\nabla_j(Y^k\partial_k) \\ &= X^j\bigl(\partial_j(Y^k)\partial_k + Y^k\nabla_j\partial_k\bigr) \\ &= X^j\bigl(\partial_j(Y^k)\partial_k + Y^k\Gamma^l_\partial_l\bigr) \\ &= X^j\bigl(\partial_j(Y^l) + Y^k\Gamma^l_\bigr)\partial_l \end that is, : (\nabla_j Y)^l = \partial_jY^l + \Gamma^l_ Y^k An affine connection \nabla is compatible with a metric iff : \partial_i \bigl(g(\partial_j, \partial_k) \bigr) = g(\nabla_i\partial_j, \partial_k) + g(\partial_j, \nabla_i\partial_k) = g(\Gamma^l_\partial_l, \partial_k) + g(\partial_j, \Gamma_^l\partial_l) i.e., if and only if : \partial_i g_ = \Gamma^l_g_ + \Gamma^l_g_. An affine connection is torsion free iff :\nabla_i\partial_j - \nabla_j \partial_i = (\Gamma^l_ - \Gamma^l_)\partial_l = partial_i, \partial_j 0. i.e., if and only if :\Gamma^l_ = \Gamma^l_ is symmetric in its lower two indices. As one checks by taking for X, Y, Z, coordinate vector fields \partial_j, \partial_k, \partial_l (or computes directly), the Koszul expression of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as :\Gamma^l_ = \tfrac g^ \left( \partial _k g_ + \partial _j g_ - \partial _r g_ \right) where as usual g^ are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix g_.


Derivative along curve

The Levi-Civita connection (like any affine connection) also defines a derivative along
curve In mathematics, a curve (also called a curved line in older texts) is an object similar to a line (geometry), line, but that does not have to be Linearity, straight. Intuitively, a curve may be thought of as the trace left by a moving point (ge ...
s, sometimes denoted by . Given a smooth curve on and a vector field along its derivative is defined by :D_tV=\nabla_V. Formally, is the pullback connection on the
pullback bundle In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in ...
. In particular, \dot\gamma(t) is a vector field along the curve itself. If \nabla_\dot(t) vanishes, the curve is called a geodesic of the covariant derivative. Formally, the condition can be restated as the vanishing of the pullback connection applied to \dot\gamma: :\left(\gamma^*\nabla\right) \dot\equiv 0. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those
geodesics In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of the
metric Metric or metrical may refer to: * Metric system, an internationally adopted decimal system of measurement * An adjective indicating relation to measurement in general, or a noun describing a specific type of measurement Mathematics In mathem ...
that are parametrised proportionally to their arc length.


Parallel transport

In general,
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 (vector bundle), c ...
along a curve with respect to a connection defines
isomorphisms In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are
orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...
– that is, they preserve the inner products on the various tangent spaces. The images below show parallel transport of the Levi-Civita connection associated to two different Riemannian metrics on the plane, expressed in
polar coordinates In mathematics, the polar coordinate system is a two-dimensional coordinate system in which each point on a plane is determined by a distance from a reference point and an angle from a reference direction. The reference point (analogous to the or ...
. The metric of left image corresponds to the standard
Euclidean metric In mathematics, the Euclidean distance between two points in Euclidean space is the length of a line segment between the two points. It can be calculated from the Cartesian coordinates of the points using the Pythagorean theorem, therefore oc ...
ds^2 = dx^2 + dy^2 = dr^2 + r^2 d\theta^2, while the metric on the right has standard form in polar coordinates (when r = 1), and thus preserves the vector tangent to the circle. This second metric has a singularity at the origin, as can be seen by expressing it in Cartesian coordinates: : dr = \frac : d\theta = \frac : dr^2 + d\theta^2 = \frac + \frac


Example: the unit sphere in

Let be the usual
scalar product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
on . Let be the
unit sphere In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit b ...
in . The tangent space to at a point is naturally identified with the vector subspace of consisting of all vectors orthogonal to . It follows that a vector field on can be seen as a map , which satisfies \bigl\langle Y(m), m\bigr\rangle = 0, \qquad \forall m\in \mathbf^2. Denote as the covariant derivative of the map in the direction of the vector . Then we have: In fact, this connection is the Levi-Civita connection for the metric on inherited from . Indeed, one can check that this connection preserves the metric.


See also

*
Weitzenböck connection Teleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In ...


Notes


References

* * See Volume I pag. 158


External links

*
MathWorld: Levi-Civita Connection



Levi-Civita connection
at the Manifold Atlas {{DEFAULTSORT:Levi-Civita Connection Riemannian geometry Connection (mathematics)