In differential geometry , the relationship is a concept that can be used to explain parallel transport between the fibers of a principal bundle . In physics, such relationships are used to describe fields in the Yang-Mills theories .
definition
Be a principal bundle with the structure group . The group works through
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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Also denote the Lie algebra of the Lie group .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
A relationship is then a -value 1-form , which is -equivariant and whose restriction to the fibers corresponds to the Maurer-Cartan form . So the following two conditions should be met:
![{\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![{\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6037692d66c7c173b726a658ad0fd3af2ff8f0ac)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
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for all
and
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for everyone .![{\ displaystyle X \ in {\ mathfrak {g}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db578a65a37022dc84959dfc2a19f694ef4b46a7)
Here is defined by . denotes the differential of . is the adjoint effect and is the so-called fundamental vector field . It will go through
![{\ displaystyle R_ {g} \ colon P \ to P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/857cf594c35617c339de11426d688c550ea16302)
![{\ displaystyle R_ {g} (p) = R (p, g)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb9b07cbe023d599a29c5ba189b13081496a61c0)
![{\ displaystyle D (R_ {g})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ff84f016c34d43fc0f03370f47d4f80d13e684b)
![R_ {g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42b87b78f0b90b93107b1f2e15c7a8961363a648)
![{\ displaystyle \ operatorname {Ad} \ colon G \ to GL ({\ mathfrak {g}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a47c004f4056266f13e48f54303b44893b210e55)
![{\ displaystyle X ^ {\ sharp}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/690e7733fa005ab72b5f48310fa9d50c6df99d7f)
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For
on defined.
![P](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a)
curvature
The curvature of a connected form is defined by
![{\ displaystyle \ Omega = d \ omega + {\ tfrac {1} {2}} [\ omega \ wedge \ omega].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68ba3d5b842315fd1628a48c741683b169f505bc)
Here the commutator is Lie algebra valued differential forms by
![{\ displaystyle [\ omega \ wedge \ eta] (v_ {1}, v_ {2}) = [\ omega (v_ {1}), \ eta (v_ {2})] - [\ omega (v_ {2 }), \ eta (v_ {1})]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a8ccefbf2bbbecdc5447b90197d94df75950cc86)
and the outer derivative through
![d \ omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/ad5fe5ef6e9587a089d370dc32119c60e91f1d4c)
![{\ displaystyle d \ omega (X, Y) = X (\ omega (Y)) - Y (\ omega (X)) - \ omega ([X, Y])}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38c3a22915a42e9a1a708716b6aa3ff84d40a7f2)
Are defined.
The curve form is -invariant and therefore defines a 2-form on .
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle \ Omega \ in \ Omega ^ {2} (M, {\ mathfrak {g}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de8891cfe56e159790cef3f0d35e890f728b61eb)
![M.](https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd)
Bianchi identity
Relationship and curvature form satisfy the equation
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Horizontal subspaces
For a form of connection on a principal bundle , the horizontal subspaces are defined by
![{\ displaystyle \ omega \ in \ Omega ^ {1} (P, {\ mathfrak {g}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6037692d66c7c173b726a658ad0fd3af2ff8f0ac)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![\ pi: P \ rightarrow M](https://wikimedia.org/api/rest_v1/media/math/render/svg/01278d9db7a95e3636ebc383271ffd95af73f9b7)
![{\ displaystyle H_ {p}, p \ in P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da1dd3b1b6c6c20812ab71992c3e706cf6dba3f2)
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.
The horizontal subspaces are transverse to the tangent spaces of the fibers of , and they are -invariant, i.e. H. for everyone .
![\pi](https://wikimedia.org/api/rest_v1/media/math/render/svg/9be4ba0bb8df3af72e90a0535fabcc17431e540a)
![G](https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b)
![{\ displaystyle H_ {gp} = DR_ {g} (H_ {p})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41078e2bca752500a450573fc7c1a8f1f94b3716)
![{\ displaystyle g \ in G, p \ in P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06fac305c539ef8f797eab7043dcb1fc0046a261)
From the horizontal subspaces can form the connection recover (for identification of the tangent of the fiber ) by projection of along the tangent of the fiber.
![{\ mathfrak {g}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40a913b1503ed9ec94361b99f7fd59ef60705c28)
![{\ displaystyle T_ {p} P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d31d930bcee4dc6d6e345708b1680653451ffae2)
![H_p](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e9dd2b2d753490bcc4be2d2d90a439caa11d72f)
Parallel transport
For every path and every there is a path with and . (This follows from the existence and uniqueness theorem for ordinary differential equations .)
![{\ displaystyle \ gamma: \ left [0.1 \ right] \ rightarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0036033ebd2c42155313562a69179fb2069b82e)
![{\ displaystyle x \ in \ pi ^ {- 1} (\ gamma (0))}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5776feb91b54cb3a2f9e84a002dd45f8b9c8bb)
![{\ displaystyle {\ tilde {\ gamma}}: \ left [0,1 \ right] \ rightarrow P}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21793d45bd3fbe48790ca0bbb5baf1b373b645f1)
![{\ displaystyle {\ tilde {\ gamma}} _ {x} (0) = x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfd971c9db7409906561aae73839a850d1ccb0a)
![{\ displaystyle \ pi ({\ tilde {\ gamma}} _ {x}) = \ gamma}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b1fc5cd1e9478a8f653eb59cdde5680a73c17dd)
In particular, you have one through
each way![{\ displaystyle \ gamma: \ left [0.1 \ right] \ rightarrow M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a0036033ebd2c42155313562a69179fb2069b82e)
![{\ displaystyle P _ {\ gamma} (x) = {\ tilde {\ gamma}} _ {x} (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/13f89f7ab1fbb25581bd13c1d5ac99147eecc8f1)
defined figure
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,
the so-called parallel transport along the way .
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
At one point , the holonomy group is defined as a subset of the diffeomorphisms of the fiber as follows. To a closed path with and one there is a clear elevation with and we define . The group that is for everyone is the holonomy group.
![b \ in B](https://wikimedia.org/api/rest_v1/media/math/render/svg/61dbfba9ff608c8700a30596649d98dcc6147d86)
![{\ displaystyle F_ {b}: = \ pi ^ {- 1} (b)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d630f9cb24d2c3afd99eec98a1e76e4c7200124)
![{\ displaystyle \ gamma: \ left [0,1 \ right] \ rightarrow B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/279b0be624cf20ca4205bd4d7e4d2f387f292586)
![{\ displaystyle \ gamma (0) = \ gamma (1) = b}](https://wikimedia.org/api/rest_v1/media/math/render/svg/39d779654b720c2c331a9c3b58ba6ab057c7d98c)
![{\ displaystyle x \ in F_ {b}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/965ac4a483ce754dcb9894fd5e0bcaa4a4d0c4cd)
![{\ tilde {\ gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a2b090b5a4f599c62fc511fa45ba936b140c7541)
![{\ displaystyle {\ tilde {\ gamma}} (0) = x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9c6d3e8ae9882272cf2da946487311d76c3dbc74)
![{\ displaystyle f _ {\ gamma} (x): = {\ tilde {\ gamma}} (1)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/44c2c6b3886e4efff47f1a07a596b01d57a47443)
![{\ displaystyle f _ {\ gamma}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8f6e7218854fd84ba91e0c3b4ee581c76d917bc)
![\gamma](https://wikimedia.org/api/rest_v1/media/math/render/svg/a223c880b0ce3da8f64ee33c4f0010beee400b1a)
Riemannian connection
For a Riemannian manifold , the frame bundle is a principal bundle with the linear group .
![GL (n, \ mathbb R)](https://wikimedia.org/api/rest_v1/media/math/render/svg/a1f7197960fac26cadfe027d3045154b9972f8d3)
Let be the matrix that by using a local base
![A.](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
![{\ displaystyle A (v) = \ nabla _ {X} v}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1267bb9ef911460079b7a2d6709690c994df2010)
is defined, where the Levi-Civita context is, so is by
![\ nabla](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3d0e93b78c50237f9ea83d027e4ebbdaef354b2)
![{\ displaystyle \ theta (X): = A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e272eb9958ee12e66eabdfbef0ed44f254702ca)
defines the Riemannian form of connection. It applies
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literature
Web links