Levi-Civita context

In mathematics , especially in Riemannian geometry , a branch of differential geometry , a Levi-Civita relationship is understood to be a connection on the tangential bundle of a Riemannian or semi-Riemannian manifold , which is in a certain way compatible with the metric of the manifold. The Levi-Civita connection plays a central role in the modern structure of Riemannian geometry. There it represents a generalization of the classical directional derivation from the multidimensional differential calculus in Euclidean spaces and is suitable for quantifying the change in direction of a vector field in the direction of another vector field. The concept of the Levi-Civita connection is equivalent to parallel transport in the sense of Levi-Civita and therefore a means to relate tangential spaces to one another at different points, which is where the term connection comes from. Since the (semi-) Riemannian geometry is an essential tool for the formulation of the general theory of relativity , the Levi-Civita connection is also used here. The Levi-Civita relationship is also used in the construction of the Dirac operator of a spin manifold .

motivation

For vector fields and in Euclidean space , the Levi-Civita relationship is defined as the directional derivative from Y to X, i.e. H. the directional derivative of the individual components from Y to X: ${\displaystyle X}$ ${\displaystyle Y=\sum _{i=1}^{n}f_{i}{\frac {\partial }{\partial x_{i}}}}$ ${\displaystyle \mathbb {R} ^{n}}$

${\displaystyle \nabla _{X}^{\mathbb {R} ^{n}}Y=\sum _{i=1}^{n}\nabla _{X}f_{i}{\frac {\partial }{\partial x_{i}}}}$,

where denotes the usual directional derivative . ${\displaystyle \nabla _{X}f_{i}}$

If is a submanifold and vector fields are on , then a vector field is defined on , but whose images are in the tangent space of , not necessarily in the tangent space of . For each one can use the orthogonal projection and then define it ${\displaystyle M\subset \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle X,Y}$ ${\displaystyle M}$ ${\displaystyle \nabla _{X}^{\mathbb {R} ^{n}}Y}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle M}$ ${\displaystyle x\in M}$ ${\displaystyle p:T_{x}\mathbb {R} ^{n}\rightarrow T_{x}M}$

${\displaystyle \nabla _{X}Y:=p(\nabla _{X}^{\mathbb {R} ^{n}}Y)}$.

This relationship fulfills the axioms given below, according to the main theorem of differential geometry it therefore agrees with the Levi-Civita relationship. The advantage of the axiomatic approach given below is that one can consider the Levi-Civita connection of a Riemann manifold independently of the embedding to be chosen . ${\displaystyle \nabla _{X}Y}$ ${\displaystyle (M,g)}$ ${\displaystyle M\rightarrow \mathbb {R} ^{n}}$

definition

Let it be a (semi-) Riemannian manifold . Then there is exactly one connection on the tangential bundle of with the following properties: ${\displaystyle (M,g)}$ ${\displaystyle \nabla }$ ${\displaystyle TM}$ ${\displaystyle M}$

• ${\displaystyle \nabla }$is torsion-free , d. H. it applies

${\displaystyle \nabla _{X}Y-\nabla _{Y}X=[X,Y]}$

for all vector fields , . Here called the Lie bracket of vector fields and . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle [X,Y]}$ ${\displaystyle X}$ ${\displaystyle Y}$

• ${\displaystyle \nabla }$is a metric relationship , i.e. H. it applies

${\displaystyle Z(g(X,Y))=g(\nabla _{Z}X,Y)+g(X,\nabla _{Z}Y)}$

for all vector fields , and . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$

This connection is called the Levi-Civita connection or also the Riemann connection of . It is named after Tullio Levi-Civita . ${\displaystyle \nabla }$ ${\displaystyle (M,g)}$

properties

Main theorem of Riemannian geometry

From the above definition it is not clear whether such a Levi-Civita connection exists at all. So this has to be proven first. The statement that such a connection exists and is also unambiguous is often called the main theorem of Riemannian geometry in the literature . The Levi-Civita connection is an essential tool for building up Riemann's theory of curvature. Because the curvature tensor is defined with the help of a relationship, it is therefore advisable to use the clearly distinguished Levi-Civita relationship for the definition of the Riemannian curvature tensor in Riemann's geometry .

Koszul formula

The Levi-Civita connection is clearly described by the Koszul formula (named after Jean-Louis Koszul ) ${\displaystyle \nabla }$

${\displaystyle {\begin{aligned}&g(\nabla _{X}Y,Z)\\&={\frac {1}{2}}{\bigl (}X(g(Y,Z))+Y(g(Z,X))-Z(g(X,Y))-g(Y,[X,Z])-g(Z,[Y,X])+g(X,[Z,Y]){\bigl )}.\end{aligned}}}$

This gives an implicit, global description of , which is particularly suitable for an abstract proof of existence of . For the construction of, however, one can also start from a local description. ${\displaystyle \nabla }$ ${\displaystyle \nabla }$ ${\displaystyle \nabla }$

Christmas symbols

A local description of can be obtained as follows. In general, a connection on a vector bundle is described locally by its connection coefficients. The connection coefficients of the Levi-Civita connection are the classic Christoffel symbols of the second kind . This means in detail that with regard to a card from ${\displaystyle \nabla }$ ${\displaystyle \Gamma _{ij}^{k}}$ ${\displaystyle (U,h)}$ ${\displaystyle M}$

${\displaystyle \nabla _{\partial _{i}}\partial _{j}=\sum _{k=1}^{n}\Gamma _{ij}^{k}\,\partial _{k}}$

With

${\displaystyle \Gamma _{ij}^{k}={\frac {1}{2}}\sum _{l=1}^{n}g^{kl}(\partial _{i}g_{jl}+\partial _{j}g_{il}-\partial _{l}g_{ij})}$

applies. Here is the inverse matrix of the Riemann fundamental tensor and the coordinate base of the map . ${\displaystyle (g^{kl})}$ ${\displaystyle (g_{kl})}$ ${\displaystyle (\partial _{1},\ldots ,\partial _{n})}$ ${\displaystyle (U,h)}$

Since the Levi-Civita relationship is torsion-free, the Christoffel symbols are symmetrical, i.e. H. for all , and the following applies: . ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle k}$ ${\displaystyle \Gamma _{ij}^{k}=\Gamma _{ji}^{k}}$

One calls the covariant derivation of along , since the classical covariant derivation from the tensor calculus of Gregorio Ricci-Curbastro and Tullio Levi-Civita generalizes. ${\displaystyle \nabla _{X}Y}$ ${\displaystyle Y}$ ${\displaystyle X}$ ${\displaystyle \nabla }$

Relationships for the derivation of direction

Let it be a (semi-) Riemannian manifold and the Levi-Civita connection of . In addition , vector fields are on . Then can be as a generalization of the concept as follows the directional derivative of vector fields of the take. ${\displaystyle (M,g)}$ ${\displaystyle \nabla }$ ${\displaystyle (M,g)}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle M}$ ${\displaystyle \nabla _{X}Y}$ ${\displaystyle \mathbb {R} ^{n}}$

• It is a point. Then it only depends on the tangent vector and the vector field . If you choose a smooth curve with and and denote the parallel transport along in the sense of Levi-Civita, then the following applies ${\displaystyle p\in M}$ ${\displaystyle \nabla _{X}Y}$ ${\displaystyle X(p)}$ ${\displaystyle Y}$ ${\displaystyle \alpha \colon (-\varepsilon ,\varepsilon )\rightarrow M}$ ${\displaystyle \alpha (0)=p}$ ${\displaystyle \alpha '(0)=X(p)}$ ${\displaystyle P^{\alpha }}$ ${\displaystyle \alpha }$

  ${\displaystyle (\nabla _{X}Y)(p)=\lim _{t\rightarrow 0}{\frac {P_{t}^{\alpha }(Y\circ \alpha )-P_{0}^{\alpha }(Y\circ \alpha )}{t}}.}$

That means, it results like the classical direction derivation as limit value of a difference quotient, whereby the “law of transplantation” ( Hermann Weyl ) from to is given by the parallel shift in the sense of Levi-Civitas. In the special case, in which the one with the standard metric, this term of a parallelism shift agrees with the conventional parallel shift im , so that in this case the usual directional derivative of a vector field along a vector field corresponds to the newly defined covariant derivative. ${\displaystyle (\nabla _{X}Y)(p)}$ ${\displaystyle T_{\alpha (0)}M}$ ${\displaystyle T_{\alpha (t)}M}$ ${\displaystyle (M,g)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$

• It is a point. Then a map exists around , so that the metric fundamental tensor in the point with respect to is given by ( normal coordinates ). With regard to such a card, point applies ${\displaystyle p\in M}$ ${\displaystyle (U,h)}$ ${\displaystyle p}$ ${\displaystyle (g_{ij})}$ ${\displaystyle p}$ ${\displaystyle (U,h)}$ ${\displaystyle (g_{ij}(p))=(\delta _{ij})}$ ${\displaystyle p}$

  ${\displaystyle (\nabla _{X}Y)(p)=\sum _{i,j=1}^{n}X_{i}(p)\left.{\frac {\partial Y_{j}}{\partial x_{i}}}(p)\partial _{j}\right|_{p},}$

if and are the local coordinates of and with respect to. I.e. with respect to normal coordinates, the local definition of is exactly the same as in the “flat case” with the standard metric. ${\displaystyle X_{i}}$ ${\displaystyle Y_{j}}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle (U,h)}$ ${\displaystyle (\nabla _{X}Y)(p)}$ ${\displaystyle \mathbb {R} ^{n}}$

The Levi-Civita connection has a particularly simple description in the case in which there is a Riemannian manifold that arises from the fact that the standard metric des is restricted to a submanifold des . In this case, the Levi-Civita connection is of given as follows. It applies ${\displaystyle (M,g)}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \nabla }$ ${\displaystyle (M,g)}$

${\displaystyle (\nabla _{X}Y)(p)=\mathrm {proj} _{T_{p}M}^{\perp }((D_{\tilde {X}}{\tilde {Y}})(p)).}$

Here, , vector fields , , continuations of these vector fields for vector fields on whole , the directional derivative of along the vector field and the orthogonal projection of the tangent space with the base point . ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle M}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle {\tilde {Y}}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle D_{\tilde {X}}{\tilde {Y}}}$ ${\displaystyle {\tilde {Y}}}$ ${\displaystyle {\tilde {X}}}$ ${\displaystyle \mathrm {proj} _{T_{p}M}^{\perp }}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle T_{p}M}$ ${\displaystyle p}$

Directional derivation along curves

The Levi-Civita connection allows the definition of the acceleration of a smooth curve that runs in a Riemannian manifold. This leads to a description of the geodesics of the underlying Riemannian manifold as the acceleration-free curves. First of all, the Levi-Civita connection (like every connection on a vector bundle) defines a directional derivation for vector fields that are explained along a curve. This directional derivative measures the rate of change of the vector field in the direction of the curve. Different terms are used for this derivation. We name the most common ones after the definition.

Let it be a smooth curve in the Riemannian manifold and a vector field along it . The directional derivative of along in the point is ${\displaystyle \alpha \colon I\rightarrow M}$ ${\displaystyle (M,g)}$ ${\displaystyle X}$ ${\displaystyle \alpha }$ ${\displaystyle X}$ ${\displaystyle \alpha }$ ${\displaystyle t_{0}\in I}$

${\displaystyle (\nabla _{\alpha }X)(t_{0}):=\nabla _{\alpha '(t_{0})}X.}$

Other common names for this size are

${\displaystyle {\frac {\nabla _{\alpha }X}{dt}}(t_{0}),\quad D_{t}X(t_{0}).}$

In particular , the velocity field of is itself a vector field along the curve . The acceleration of is along the vector field . The curve is a geodesic of the Riemannian manifold if and only if its acceleration vanishes. From a physical standpoint, geodesics can be interpreted kinematically as the curves that a particle in the Riemannian manifold would follow if it is not exposed to any force. ${\displaystyle \alpha '}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle \nabla _{\alpha '}\alpha '}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$ ${\displaystyle (M,g)}$

Parallel transport

In general, a parallel transport along a curve defines an isomorphism between the fibers whose base points lie on the curve with respect to a relationship on a vector bundle. If the connection is the Levi-Civita connection of a Riemannian manifold, then the isomorphisms are orthogonal , i.e. true to length and angle. The parallel transport induced by the Levi-Civita connection of a Riemannian manifold agrees with the parallel transport first defined by Levi-Civita in 1918 (cf. parallel transport in the sense of Levi-Civita ). This was anticipated in a special case by Ferdinand Minding .

Riemannian connection

In the theory of the principal bundle , relationships are defined as Lie algebra-valued 1-forms. Since the frame bundle of a Riemannian manifold is a principal bundle with the general linear group as a structural group , one can define a connection form with the help of the Levi-Civita connection as follows. ${\displaystyle \pi \colon F(M)\to M}$ ${\displaystyle M}$ ${\displaystyle GL(n,\mathbb {R} )}$ ${\displaystyle \nabla }$

Let local coordinates be in a neighborhood of such that the base ${\displaystyle (x_{i})_{i}}$ ${\displaystyle p\in M}$

${\displaystyle B=\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right)}$

is an element of the frame bundle, so . The Christoffel symbols of the Levi-Civita connection are then through ${\displaystyle B\in F(M)_{p}}$ ${\displaystyle \Gamma _{ij}^{k}}$

${\displaystyle \nabla _{\frac {\partial }{\partial x_{i}}}{\frac {\partial }{\partial x_{j}}}=\sum _{k=1}^{n}\Gamma _{ij}^{k}{\frac {\partial }{\partial x_{k}}}}$

described. By defined valent-1 form to have in these coordinates the decomposition ${\displaystyle \theta (X):=B^{-1}(\pi (X))}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle F(M)}$

${\displaystyle \theta (X)=\sum _{i=1}^{n}\theta _{ji}{\frac {\partial }{\partial x_{i}}}}$.

Be

${\displaystyle (X_{1},\ldots ,X_{n})=\left({\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right)}$

the basis continued on an environment of . Then defined ${\displaystyle T_{p}M}$

${\displaystyle (\omega _{ij})_{ij}=\left(\sum _{k=1}^{n}\theta _{ik}\left(dX_{jk}+\sum _{l,m}\Gamma _{ml}^{k}X_{jl}dx_{m}\right)\right)_{ij}}$

a matrix-valued 1-form and it holds

${\displaystyle \theta \in \Omega ^{1}(F(M),{\mathfrak {gl}}(n,\mathbb {R} )).}$

The parallel transport on the frame bundle defined by the Riemann connection corresponds to the parallel transport on the tangential bundle defined by the Levi-Civita connection.

literature

• Isaac Chavel : Riemannian Geometry. A Modern Introduction. 2nd Edition. Cambridge University Press, Cambridge 2006, ISBN 0521619548 .
• John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228 .
• Barrett O'Neill: Semi-Riemannian Geometry. With Applications to Relativity. Academic Press, New York 1983, ISBN 0125267401 .
• Michael Spivak: A Comprehensive Introduction to Differential Geometry (Volume 2). Publish or Perish Press, Berkeley 1999, ISBN 0-914098-71-3 .
• Rainer Oloff: Geometry of space-time. A mathematical introduction to the theory of relativity. 3. Edition. Vieweg, Wiesbaden 2004, ISBN 3528269170 .
• Hermann Weyl : Space, Time, Matter. Springer, 1923.

Web links

• MathWorld: Levi-Civita Connection
• PlanetMath: Levi-Civita Connection

Individual evidence

1. ↑ Kobayashi, Nomizu: Foundations of Differential Geometry, Section iii.7