Torsion tensor

The torsion tensor is a mathematical object from the field of differential geometry . This tensor field was introduced by Élie Cartan in his studies on geometry and gravitation .

definition

Let be a differentiable manifold together with an affine connection . The torsion tensor is a tensor field that goes through ${\displaystyle (M,\nabla )}$ ${\displaystyle \nabla }$ ${\displaystyle T}$

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

is defined. There are two vector fields and means the Lie bracket . ${\displaystyle X,Y\in \Gamma (TM)}$ ${\displaystyle [\cdot ,\cdot ]}$

Local representation

Let be a local frame of the tangent bundle . These are cuts in the tangent bundle that form a vector space basis in every tangent space . If one sets , and , then holds for the components of the torsion tensor in local coordinates ${\displaystyle e_{1},\ldots ,e_{n}}$ ${\displaystyle TM}$ ${\displaystyle X:=e_{i}}$ ${\displaystyle Y:=e_{j}}$ ${\displaystyle \gamma _{ij}^{k}e_{k}:=[e_{i},e_{j}]}$ ${\displaystyle T_{ij}^{k}}$

${\displaystyle T^{k}{}_{ij}=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji}-\gamma ^{k}{}_{ij},\quad i,j,k=1,2,\ldots ,n.}$

The symbols denote the Christoffel symbols . Since it is always possible to choose the local frame so that the Lie bracket disappears everywhere, the following applies to the components of the tensor field in these coordinates ${\displaystyle \Gamma _{ij}^{k}}$

${\displaystyle T^{k}{}_{ij}=\Gamma ^{k}{}_{ij}-\Gamma ^{k}{}_{ji},\quad i,j,k=1,2,\ldots ,n.}$

properties

• The torsion tensor is a (2,1) -tensor field, so it is in particular - linear in its three arguments. ${\displaystyle C^{\infty }}$
• The torsion tensor is skew symmetric, that is, it holds . ${\displaystyle T(X,Y)=-T(Y,X)}$

Symmetrical connection

An affine relationship is called symmetric or torsion-free if the torsion tensor vanishes, i.e. if ${\displaystyle \nabla }$

${\displaystyle T(X,Y)=0}$

or equivalent

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

applies. The most important symmetrical relationship is the Levi-Civita relationship , which is also metric .

For symmetrical relationships, a kind of generalization of Schwarz's theorem for differentiable curves can be proven. Let be a differentiable manifold with a symmetrical connection and a smooth homotopy of smooth curves , then we have ${\displaystyle M}$ ${\displaystyle \nabla }$ ${\displaystyle c\colon \left]-\epsilon ,\epsilon \right[\times \left]a,b\right[\to M}$

${\displaystyle \nabla _{\frac {\partial }{\partial s}}{\frac {\partial }{\partial t}}c(s,t)=\nabla _{\frac {\partial }{\partial t}}{\frac {\partial }{\partial s}}c(s,t)\,.}$

Simply stated, therefore, the derivative may by in the case of a symmetrical relationship with the to be exchanged. ${\displaystyle s}$ ${\displaystyle t}$

literature

• Torsion tensor . In: Michiel Hazewinkel (Ed.): Encyclopaedia of Mathematics . Springer-Verlag , Berlin 2002, ISBN 978-1-55608-010-4 (English, online ).

Individual evidence

1. ↑ Elie Cartan: On manifolds with an Affine Connection and the Theory of General Relativity (= Monographs and Textbooks in Physical Science 1). Bibliopolis, Neapol 1986, ISBN 88-7088-086-9 (Engl. Transl. Of French original 1922/23: Sur les variétés à connexion affine et la théorie de la relativité généralisée ).
2. ↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , p. 68.
3. ↑ John M. Lee: Riemannian Manifolds. An Introduction to Curvature (= Graduate Texts in Mathematics 176). Springer, New York NY et al. 1997, ISBN 0-387-98322-8 , pp. 97-98.