Curvature form

 ( Curvature Tensors ) 

In differential geometry, the curvature form describes curvature of a connection form 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 $\backslash mathfrak\; g$, and P → B be a principal bundle. Let ω be an Ehresmann connection on P (which is a $\backslash mathfrak\; g$-valued one-form on P).

Then the curvature form is the $\backslash mathfrak\; g$-valued 2-form on P defined by

$\backslash Omega=d\backslash omega\; +\; \{1\; \backslash over\; 2\}\backslash omega\; =\; D\; \backslash omega.$

(In another convention, 1/2 does not appear.) Here $d$ stands for exterior derivative, $\backslash cdot$ is defined in the article "Lie algebra-valued form" and D denotes the exterior covariant derivative. In other terms,since $\backslash omega(X,\; Y)\; =\; \backslash frac\{1\}\{2\}(\backslash omega(X),\; -\; \backslash omega(Y),)$. Here we use also the $\backslash sigma=2$ Kobayashi convention for the exterior derivative of a one form which is then $d\backslash omega(X,\; Y)\; =\; \backslash frac12(X\backslash omega(Y)\; -\; Y\; \backslash omega(X)\; -\; \backslash omega(X,))$

$\backslash ,\backslash Omega(X,\; Y)=\; d\backslash omega(X,Y)\; +\; \{1\; \backslash over\; 2\}\backslash omega(X),\backslash omega(Y)$

where X, Y are tangent vectors to P.

There is also another expression for Ω: if X, Y are horizontal vector fields on P, thenProof: $\backslash sigma\backslash Omega(X,\; Y)\; =\; \backslash sigma\; d\backslash omega(X,\; Y)\; =\; X\backslash omega(Y)\; -\; Y\; \backslash omega(X)\; -\; \backslash omega(X,)\; =\; -\backslash omega(X,).$

$\backslash sigma\backslash Omega(X,\; Y)\; =\; -\backslash omega(X,)\; =\; -X,\; +\; hX,$

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 $\backslash sigma\backslash in\; \backslash \{1,\; 2\backslash \}$ 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:

$\backslash ,\backslash Omega\; =\; d\backslash omega\; +\; \backslash omega\; \backslash wedge\; \backslash omega,$

where $\backslash wedge$ is the exterior power. More precisely, if $\{\backslash omega^i\}\_j$ and $\{\backslash Omega^i\}\_j$ denote components of ω and Ω correspondingly, (so each $\{\backslash omega^i\}\_j$ is a usual 1-form and each $\{\backslash Omega^i\}\_j$ is a usual 2-form) then

$\backslash Omega^i\_j\; =\; d\{\backslash omega^i\}\_j\; +\; \backslash sum\_k\; \{\backslash omega^i\}\_k\; \backslash wedge\; \{\backslash omega^k\}\_j.$

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.

$\backslash ,R(X,\; Y)\; =\; \backslash Omega(X,\; Y),$

using the standard notation for the Riemannian curvature tensor.

---

Bianchi identities If $\backslash theta$ is the canonical vector-valued 1-form on the frame bundle, the torsion $\backslash Theta$ of the connection form $\backslash omega$ is the vector-valued 2-form defined by the structure equation

$\backslash Theta\; =\; d\backslash theta\; +\; \backslash omega\backslash wedge\backslash theta\; =\; D\backslash theta,$

where as above D denotes the exterior covariant derivative.

The first Bianchi identity takes the form

$D\backslash Theta\; =\; \backslash Omega\backslash wedge\backslash theta.$

The second Bianchi identity takes the form

$\backslash ,\; D\; \backslash Omega\; =\; 0$

and is valid more generally for any Connection form in a principal bundle.

The Bianchi identities can be written in tensor notation as: $R\_\{abmn;\backslash ell\}\; +\; R\_\{ab\backslash ell\; m;n\}\; +\; R\_\{abn\backslash ell;m\}\; =\; 0.$

The contracted Bianchi identities are used to derive the Einstein tensor in the Einstein field equations, a key component in the general theory of relativity.

---

Notes

• 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

Post Comment