In
Riemannian geometryRiemannian 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 which varies smoothly from point to point. This gives, in particular, local notions of angle, length...
, the
Levi-Civita connection is a specific
connectionIn geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
on the tangent bundle of a
manifoldIn mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. More specifically, it is the torsion-free
metric connectionIn mathematics, a metric connection is a connection in a vector bundle E equipped with a metric for which the inner product of any two vectors will remain the same when those vectors are parallel transported along any curve...
, i.e., the torsion-free
connectionIn geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner. There are a variety of kinds of connections in modern geometry, depending on what sort of data one wants to transport...
on the
tangent bundleIn differential geometry, the tangent bundle of a differentiable manifold M is the disjoint unionThe disjoint union assures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector...
(an
affine connectionIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
) preserving a given (
pseudo-In differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
)Riemannian metric.
The
fundamental theorem of Riemannian geometryIn Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric...
states that there is a unique connection which satisfies these properties.
In the theory of
RiemannianIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
and
pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
s the term
covariant derivativeIn 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 differential operator, to be contrasted with the approach given...
is often used for the Levi-Civita connection. The components of this connection with respect to a system of local coordinates are called
Christoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
.
The Levi-Civita connection is named after
Tullio Levi-CivitaTullio Levi-Civita, FRS was an Italian mathematician, most famous for his work on absolute differential calculus and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus...
, although originally "discovered" by
Elwin Bruno ChristoffelElwin Bruno Christoffel was a German mathematician and physicist.-Life:...
. Levi-Civita, along with
Gregorio Ricci-CurbastroGregorio Ricci-Curbastro was an Italian mathematician. He was born at Lugo di Romagna. He is most famous as the inventor of the tensor calculus but published important work in many fields....
, used Christoffel's symbols
to define the notion of
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
and explore the relationship of parallel transport with the
curvatureIn the mathematical field of differential geometry, the Riemann curvature tensor, or Riemann–Christoffel tensor after Bernhard Riemann and Elwin Bruno Christoffel, is the most standard way to express curvature of Riemannian manifolds...
, thus developing the modern notion of
holonomyIn 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 geometrical data being transported. For flat connections,...
.
The Levi-Civita notions of intrinsic derivative 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
, since the definition of the Christoffel symbols
make sense in any Riemannian manifold. In 1869, Christoffel discovered that the components
of the intrinsic derivative of a vector transform as the components of a contravariant vector.
This discovery was the real beginning of tensor analysis. It was not until 1917 that Levi-Civita
interpreted the intrinsic derivative in the case of an embedded surface as the tangential component
of the usual derivative in the ambient affine space.
Formal definition
Let
be a
Riemannian manifoldIn Riemannian geometry and the differential geometry of surfaces, a Riemannian manifold or Riemannian space is a real differentiable manifold M in which each tangent space is equipped with an inner product g, a Riemannian metric, which varies smoothly from point to point...
(or
pseudo-Riemannian manifoldIn differential geometry, a pseudo-Riemannian manifold is a generalization of a Riemannian manifold. It is one of many mathematical objects named after Bernhard Riemann. The key difference between a Riemannian manifold and a pseudo-Riemannian manifold is that on a pseudo-Riemannian manifold the...
).
Then an
affine connectionIn the branch of mathematics called differential geometry, an affine connection is a geometrical object on a smooth manifold which connects nearby tangent spaces, and so permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space...
is called a Levi-Civita connection if
- it preserves the metric, i.e., .
- it is torsion-free, i.e., for any vector fields and we have , where is the Lie bracket
In mathematics, the Lie derivative , named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a vector field or more generally a tensor field, along the flow of another vector field...
of the vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
s and .
Condition 1 above is sometimes referred to as compatibility with the metric,
and condition 2 is sometimes called symmetry, cf. DoCarmo's text.
Assuming a Levi Civita connection exists it is uniquely determined. Using conditions 1 and the symmetry of the metric tensor
we find:
By condition 2 the right hand side is equal to
so we find
Since
is arbitrary, this uniquely determines
. Conversely, using the last line as a definition one shows that the expression so defined is a connection compatible with the metric, i.e. is a Levi Civita connection.
Christoffel symbols
Let
be the connection of the Riemannian metric. Choose local coordinates
and let
be the
Christoffel symbolsIn mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel , are numerical arrays of real numbers that describe, in coordinates, the effects of parallel transport in curved surfaces and, more generally, manifolds. As such, they are coordinate-space expressions for the...
with respect to these coordinates. The torsion freeness condition 2 is then equivalent to the symmetry
The definition of the Levi Civita connection derived above is equivalent to a definition of the
Christoffel symbols in terms of the metric as
where as usual
are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix
.
Derivative along curve
The Levi-Civita connection (like any affine connection) also defines a derivative along
curveIn mathematics, a curve is, generally speaking, an object similar to a line but which is not required to be straight...
s, sometimes denoted by
.
Given a smooth curve
on
and a
vector fieldIn vector calculus, a vector field is an assignmentof a vector to each point in a subset of Euclidean space. A vector field in the plane for instance can be visualized as an arrow, with a given magnitude and direction, attached to each point in the plane...
along
its derivative is defined by
(Formally
D is the pullback connection on the
pullback bundleIn mathematics, a pullback bundle or induced bundle is a useful construction in the theory of fiber bundles. Given a fiber bundle π : E → B and a continuous map f : B′ → B one can define a "pullback" of E by f as a bundle f*E over B′...
γ*T
M.)
In particular,
is a vector field along the curve
itself. If
vanishes, the curve is called a geodesic of the covariant derivative. If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those geodesics of the
metricIn the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
that are parametrised proportionally to their arc length.
Parallel transport
In general,
parallel transportIn geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection , then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the...
along a curve with respect to a connection defines isomorphisms between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are
orthogonalIn mathematics, the orthogonal group of degree n over a field F is the group of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication...
– that is, they preserve the inner products on the various tangent spaces.
The unit sphere in
Let
be the usual scalar product on
.
Let
be the unit sphere in
. The tangent space to
at a point
is naturally identified with the vector sub-space of
consisting of all vectors orthogonal to
. It follows that a vector field
on
can be seen as a map
which satisfies
Denote by
the differential of such a map. Then we have:
Lemma
The formula
defines an affine connection on
with vanishing torsion.
Proof
It is straightforward to prove that
satisfies the Leibniz identity and is
linear in the first variable. It is also a straightforward computation to show that this connection is torsion free.
So all that needs to be proved here is that the formula above does indeed define a vector field. That is, we need to prove that for all
in
Consider the map
The map
is constant, hence its differential vanishes. In particular
The equation (1) above follows.
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.
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