Английская Википедия:Curvature form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let G be a Lie group with Lie algebra <math>\mathfrak g</math>, and P → B be a principal G-bundle. Let ω be an Ehresmann connection on P (which is a <math>\mathfrak g</math>-valued one-form on P).
Then the curvature form is the <math>\mathfrak g</math>-valued 2-form on P defined by
- <math>\Omega=d\omega + {1 \over 2}[\omega \wedge \omega] = D \omega.</math>
(In another convention, 1/2 does not appear.) Here <math>d</math> stands for exterior derivative, <math>[\cdot \wedge \cdot]</math> is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,[1]
- <math>\,\Omega(X, Y)= d\omega(X,Y) + {1 \over 2}[\omega(X),\omega(Y)]</math>
where X, Y are tangent vectors to P.
There is also another expression for Ω: if X, Y are horizontal vector fields on P, then[2]
- <math>\sigma\Omega(X, Y) = -\omega([X, Y]) = -[X, Y] + h[X, Y]</math>
where hZ means the horizontal component of Z, on the right we identified a vertical vector field and a Lie algebra element generating it (fundamental vector field), and <math>\sigma\in \{1, 2\}</math> is the inverse of the normalization factor used by convention in the formula for the exterior derivative.
A connection is said to be flat if its curvature vanishes: Ω = 0. Equivalently, a connection is flat if the structure group can be reduced to the same underlying group but with the discrete topology.
Curvature form in a vector bundle
If E → B is a vector bundle, then one can also think of ω as a matrix of 1-forms and the above formula becomes the structure equation of E. Cartan:
- <math>\,\Omega = d\omega + \omega \wedge \omega, </math>
where <math>\wedge</math> is the wedge product. More precisely, if <math>{\omega^i}_j</math> and <math>{\Omega^i}_j</math> denote components of ω and Ω correspondingly, (so each <math>{\omega^i}_j</math> is a usual 1-form and each <math>{\Omega^i}_j</math> is a usual 2-form) then
- <math>\Omega^i_j = d{\omega^i}_j + \sum_k {\omega^i}_k \wedge {\omega^k}_j.</math>
For example, for the tangent bundle of a Riemannian manifold, the structure group is O(n) and Ω is a 2-form with values in the Lie algebra of O(n), i.e. the antisymmetric matrices. In this case the form Ω is an alternative description of the curvature tensor, i.e.
- <math>\,R(X, Y) = \Omega(X, Y),</math>
using the standard notation for the Riemannian curvature tensor.
Bianchi identities
Шаблон:See also Шаблон:See also If <math>\theta</math> is the canonical vector-valued 1-form on the frame bundle, the torsion <math>\Theta</math> of the connection form <math>\omega</math> is the vector-valued 2-form defined by the structure equation
- <math>\Theta = d\theta + \omega\wedge\theta = D\theta,</math>
where as above D denotes the exterior covariant derivative.
The first Bianchi identity takes the form
- <math>D\Theta = \Omega\wedge\theta.</math>
The second Bianchi identity takes the form
- <math>\, D \Omega = 0 </math>
and is valid more generally for any connection in a principal bundle.
The Bianchi identities can be written in tensor notation as: <math> R_{abmn;\ell} + R_{ab\ell m;n} + R_{abn\ell;m} = 0.</math>
The contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, the bulk of general theory of relativity.Шаблон:Clarify
Notes
References
- Shoshichi Kobayashi and Katsumi Nomizu (1963) Foundations of Differential Geometry, Vol.I, Chapter 2.5 Curvature form and structure equation, p 75, Wiley Interscience.
See also
- Connection (principal bundle)
- Basic introduction to the mathematics of curved spacetime
- Contracted Bianchi identities
- Einstein tensor
- Einstein field equations
- General theory of relativity
- Chern-Simons form
- Curvature of Riemannian manifolds
- Gauge theory
- ↑ since <math>[\omega \wedge \omega](X, Y) = \frac{1}{2}([\omega(X), \omega(Y)] - [\omega(Y), \omega(X)])</math>. Here we use also the <math>\sigma=2 </math> Kobayashi convention for the exterior derivative of a one form which is then <math>d\omega(X, Y) = \frac12(X\omega(Y) - Y \omega(X) - \omega([X, Y])) </math>
- ↑ Proof: <math>\sigma\Omega(X, Y) = \sigma d\omega(X, Y) = X\omega(Y) - Y \omega(X) - \omega([X, Y]) = -\omega([X, Y]).</math>