Riemann curvature tensor

The Riemann curvature tensor (also called Riemann tensor , Riemannian curvature or curvature tensor ) describes the curvature of spaces of any dimension, more precisely Riemannian or pseudo-Riemannian manifolds . It was named after the mathematician Bernhard Riemann and is one of the most important tools of Riemannian geometry . It finds another important application in connection with the curvature of space-time in general relativity .

The Riemann curvature tensor is a level 4 tensor . For example, its coefficients can be given in the form . Einstein's summation convention is used in this article . ${\displaystyle R_{ikp}^{m}}$

motivation

Diffeomorphisms are the structure-preserving mappings between differentiable manifolds and accordingly (smooth) isometries are the structure-preserving mappings between Riemannian manifolds . Since differentiable manifolds are by definition locally diffeomorphic to Euclidean space , the question arose whether Riemannian manifolds are also locally isometric to . This is not the case. Therefore the Riemannian curvature tensor was introduced, which, simply put, indicates how locally similar a Riemannian manifold is. In order to better understand the definition of the Riemann curvature tensor, the following consideration is introduced in. ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle \mathbb {R} ^{2}}$

Be a vector field. In Euclidean equality applies to the unit vector fields along the coordinate axes ${\displaystyle Z\in \Gamma (T\mathbb {R} ^{2})}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle \partial _{1},\,\partial _{2}}$

${\displaystyle \nabla _{\partial _{1}}\nabla _{\partial _{2}}Z=\nabla _{\partial _{2}}\nabla _{\partial _{1}}Z,}$

which Black's theorem secures. This no longer applies to general vector fields either. If I have the representation in coordinates , then applies ${\displaystyle X,\ Y}$ ${\displaystyle \mathbb {R} ^{2}}$ ${\displaystyle Z}$ ${\displaystyle \textstyle Z=Z^{i}\partial _{i}}$

${\displaystyle \nabla _{X}\nabla _{Y}Z=\nabla _{X}YZ^{i}\partial _{i}=XYZ^{i}\partial _{i}.}$

The term denotes the directional derivative of in direction . If one now investigates further the non-commutativity of , one obtains in Euclidean space ${\displaystyle YZ^{i}}$ ${\displaystyle Z^{i}}$ ${\displaystyle Y}$ ${\displaystyle \nabla _{X}\nabla _{Y}}$

${\displaystyle \nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z=(XYZ^{i}-YXZ^{i})\partial _{i}=\nabla _{[X,Y]}Z.}$

This is wrong on general manifolds. For this reason, the following definition is made.

definition

Be a smooth manifold with the context . Then the Riemann curvature tensor is a map ${\displaystyle M}$ ${\displaystyle \nabla }$

${\displaystyle \Gamma ^{\infty }(M,TM)\times \Gamma ^{\infty }(M,TM)\times \Gamma ^{\infty }(M,TM)\to \Gamma ^{\infty }(M,TM),}$

which through

${\displaystyle R(X,Y)Z=\nabla _{X}\nabla _{Y}Z-\nabla _{Y}\nabla _{X}Z-\nabla _{[X,Y]}Z}$

is defined. With is meant the space of the smooth vector fields and with the Lie bracket . ${\displaystyle \Gamma ^{\infty }(M,TM)}$ ${\displaystyle [.,.]}$

The curvature tensor can be represented in local coordinates with the help of the Christoffels symbols:

${\displaystyle R_{ikp}^{m}=\partial _{k}\Gamma _{ip}^{m}-\partial _{p}\Gamma _{ik}^{m}+\Gamma _{ip}^{a}\Gamma _{ak}^{m}-\Gamma _{ik}^{a}\Gamma _{ap}^{m}}$

annotation

Some authors, such as do Carmo or Gallot, Hulin, Lafontaine, define the Riemann curvature tensor with the opposite sign . In this case, the sign in the definition of the section curvature and the Ricci curvature also rotates, so that the signs of the section curvature, Ricci curvature and scalar curvature match for all authors.

properties

Tensor field

The curvature tensor is a - tensor field . ${\displaystyle (1,3)}$

• In particular, it is linear in every variable .
• The value of at the point of the manifold depends only on the values ​​of the vector fields , and at the point (and not on the values ​​in a neighborhood of .) ${\displaystyle R(X,Y)Z}$ ${\displaystyle p}$ ${\displaystyle X}$ ${\displaystyle Y}$ ${\displaystyle Z}$ ${\displaystyle p}$ ${\displaystyle p}$

Symmetries of the curvature tensor

On a differentiable manifold with any connection, the curvature tensor is skew-symmetric in the first two entries, that is, it holds ${\displaystyle M}$

• ${\displaystyle R(X,Y)Z=-R(Y,X)Z.}$

For Riemannian manifolds with the Levi-Civita connection also applies ${\displaystyle (M,g)}$

• ${\displaystyle g(R(X,Y)Z,T)=-g(R(X,Y)T,Z)}$ and
• ${\displaystyle g(R(X,Y)Z,T)=g(R(Z,T)X,Y)}$ ("Block swap").

Bianchi identities

If there is a differentiable manifold with connection and are vector fields, then the first Bianchi identity holds ${\displaystyle M}$ ${\displaystyle \nabla }$ ${\displaystyle W,X,Y,Z\in \Gamma ^{\infty }(M,TM)}$

• ${\displaystyle R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=(\nabla _{X}T)(Y,Z)+T(T(X,Y),Z)+(\nabla _{Y}T)(Z,X)+T(T(Y,Z),X)+(\nabla _{Z}T)(X,Y)+T(T(Z,X),Y)}$

with the torsion tensor and ${\displaystyle T}$ ${\displaystyle (\nabla _{X}T)(Y,Z)=\nabla _{X}(T(Y,Z))-T(\nabla _{X}Y,Z)-T(Y,\nabla _{X}Z).}$

The second Bianchi identity is

• ${\displaystyle (\nabla _{X}R)(Y,Z)+R(T(X,Y),Z)+(\nabla _{Y}R)(Z,X)+R(T(Y,Z),X)+(\nabla _{Z}R)(X,Y)+R(T(Z,X),Y)=0}$

With ${\displaystyle (\nabla _{X}R)(Y,Z)W=\nabla _{X}(R(Y,Z)W)-R(\nabla _{X}Y,Z)W-R(Y,\nabla _{X}Z)W-R(Y,Z)\nabla _{X}W.}$

If torsion-free, these equations simplify to ${\displaystyle \nabla }$

• ${\displaystyle R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0}$

and

• ${\displaystyle (\nabla _{X}R)(Y,Z)+(\nabla _{Y}R)(Z,X)+(\nabla _{Z}R)(X,Y)=0.}$

If there is a Riemannian manifold with the Levi-Civita connection , then the first Bianchi identity applies ${\displaystyle (M,g)}$ ${\displaystyle \nabla }$

• ${\displaystyle R(X,Y)Z+R(Y,Z)X+R(Z,X)Y=0}$

and the second Bianchi identity can be described as

• ${\displaystyle \nabla _{W}g(R(X,Y)Z,V)+\nabla _{Z}g(R(X,Y)V,W)+\nabla _{V}g(R(X,Y)W,Z)=0}$

write. The first Bianchi identity is also called the algebraic Bianchi identity and the second is also called the differential Bianchi identity. These identities are named after the mathematician Luigi Bianchi .

Flat manifold

→ Main article : Flat manifold

definition

A Riemannian manifold is called flat if it is locally isometric to Euclidean space. This means that for each point there is an environment and a mapping which is isometric, i.e. which applies to which . Here denotes the Euclidean scalar product and the push forward of . ${\displaystyle (M,g)}$ ${\displaystyle p\in M}$ ${\displaystyle U}$ ${\displaystyle \phi \colon U\to V\subset \mathbb {R} ^{n}}$ ${\displaystyle g(X,Y)=\phi ^{*}{\overline {g}}(X,Y)={\overline {g}}(\phi _{*}X,\phi _{*}Y)}$ ${\displaystyle {\overline {g}}}$ ${\displaystyle \phi _{*}}$ ${\displaystyle \phi }$

Connection to the curvature tensor

A Riemannian manifold with a Levi-Civita connection is flat if and only if the Riemannian curvature tensor is identically zero. Hence the developable surface is the two-dimensional analogue of the flat manifold. ${\displaystyle \nabla }$

Derived quantities

Cutting curvature

→ Main article : Section curvature

One of the most important curvature quantities in Riemannian geometry is the cutting curvature. It generalizes the Gaussian curvature of regular surfaces . Each plane in the tangential space is assigned a curvature at a point of a Riemannian manifold . This is the Gaussian curvature of a surface in which has a tangential plane and is not curved within the manifold, so to speak a “cut” through the manifold in the direction of the plane . The definition does not take place with the help of this surface, but with the help of the Riemann curvature tensor and two vectors that span the plane . ${\displaystyle \sigma }$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$ ${\displaystyle \sigma }$

Consider a Riemannian manifold with a Riemannian metric , a point in and a two-dimensional subspace (plane) of the tangential space of in the point . Let be and two tangent vectors that span this plane. With ${\displaystyle M}$ ${\displaystyle g}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle \sigma \subset T_{p}M}$ ${\displaystyle T_{p}M}$ ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle v}$ ${\displaystyle w}$

${\displaystyle |v\wedge w|={\sqrt {g(v,v)g(w,w)-g(v,w)^{2}}}}$

is the area of ​​the parallelogram spanned by and . Then the size depends ${\displaystyle v}$ ${\displaystyle w}$

${\displaystyle K(v,w)={\frac {g(R(v,w)w,v)}{|v\wedge w|^{2}}}={\frac {g(R(v,w)w,v)}{g(v,v)g(w,w)-g(v,w)^{2}}}}$

only on the plane , but not on the choice of vectors and spanning it . Therefore one writes for also and calls this the cutting curvature of . ${\displaystyle \sigma }$ ${\displaystyle v}$ ${\displaystyle w}$ ${\displaystyle K(v,w)}$ ${\displaystyle K(\sigma )}$ ${\displaystyle \sigma }$

If two-dimensional, then in every point of there is only one such two-dimensional subspace of the tangent space, namely the tangent space itself, and then it is precisely the Gaussian curvature of in the point ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M}$ ${\displaystyle K(\sigma )}$ ${\displaystyle M}$ ${\displaystyle p}$

Ricci tensor

In the Einstein field equations is Ricci tensor (after Gregorio Ricci-Curbastro ) was used. It results from the curvature tensor through tensor taper : ${\displaystyle R_{\mu \nu }}$

${\displaystyle R_{\mu \nu }=\pm R_{\mu \lambda \nu }^{\lambda }}$

In accordance with Einstein's summation convention , identical indices are added, one of which is at the top and the other at the bottom. To form the Ricci tensor, the index is added up. The sign is determined by convention and can in principle be freely selected. ${\displaystyle \lambda }$

Scalar curvature

The tensor taper or contraction of the Ricci tensor is called the curvature scalar (also Ricci scalar or scalar curvature ). To describe its shape, the expression is first derived from the Ricci tensor: ${\displaystyle R_{\kappa }^{\lambda }}$

${\displaystyle R_{\kappa }^{\lambda }=g^{\mu \lambda }R_{\mu \kappa }.}$

Where is the contravariant metric tensor . The curvature scalar results from contraction, with the index being added up. ${\displaystyle g^{\mu \lambda }}$ ${\displaystyle \lambda }$

${\displaystyle R=R_{\lambda }^{\lambda }}$

The curvature scalar can also be obtained directly from the Ricci tensor : ${\displaystyle R_{\mu \rho }}$

${\displaystyle R=g^{\mu \rho }R_{\mu \rho }}$

The indices and are added up. ${\displaystyle \mu }$ ${\displaystyle \rho }$

In general relativity , the curvature scalar is related to the Laue scalar via the Einstein factor , which is formed by contraction from the energy-momentum tensor : ${\displaystyle \kappa }$ ${\displaystyle T}$ ${\displaystyle T_{\nu }^{\mu }}$

${\displaystyle T=T_{\lambda }^{\lambda }=R/\kappa }$

Individual evidence

1. ^ Manfredo Perdigão do Carmo: Riemannian Geometry. 1992, p. 89
2. ^ Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine: Riemannian Geometry. 2nd edition 1990, p. 107

literature

• Manfredo Perdigão do Carmo: Riemannian Geometry , Birkhäuser, Boston 1992, ISBN 0-8176-3490-8
• Sylvestre Gallot, Dominique Hulin, Jacques Lafontaine: Riemannian Geometry. 2nd Edition. Springer-Verlag, Berlin / Heidelberg 1990, ISBN 3-540-52401-0
• John M. Lee: Riemannian Manifolds. An Introduction to Curvature. Springer, New York 1997, ISBN 0387983228 .
• Peter W. Michor: Topics in Differential Geometry. AMS, Providence, RI, 2008, ISBN 978-0-8218-2003-2

Web links

• Riemann curvature tensor on MathWorld
• Video: Riemann curvature tensor . Jörn Loviscach 2014, made available by the Technical Information Library (TIB), doi : 10.5446 / 19927 .