
In differential geometry, the holonomy group of a connection (mathematics) on a vector bundle over a smooth manifold ''M'' is the group (mathematics) of linear transformations induced by parallel transport around closed loops in ''M''. There is an analogous notion for connection form on principal bundles over ''M''. The holonomy group of a connection is intimately associated with the curvature form of that connection. The holonomy group of a Riemannian manifold ''M'' is the just holonomy group of the Levi-Civita connection on the tangent bundle of ''M''. ==On vector bundles== Let ''E'' be a rank ''k'' vector bundle over a smooth manifold ''M'' and let ∇ be a connection (mathematics) on ''E''. Given a piecewise smooth loop (topology) γ : [0,1] → ''M'' based at ''x'' in ''M'', the connection defines a parallel transport map . This map is both linear and invertible and so defines an element of GL(''Ex''). The holonomy group of ∇ based at ''x'' is defined as : The local holonomy group based at ''x'' is the subgroup coming from contractible loops γ. If ''M'' is connected space then the holonomy group depends on the basepoint ''x'' only up to conjugation in GL(''k'', R). Explicitly, if γ is a path from ''x'' to ''y'' in ''M'' then : Choosing different identifications of ''Ex'' with R''k'' also gives conjugate subgroups. It is therefore customary to drop reference to the basepoint with the understanding that the definition is good up to conjugation. Some important properties of holonomy group include: *Hol0(∇) is a connected, Lie subgroup of GL(''k'', R). *Hol0(∇) is the identity component of Hol(∇). *There is a natural, surjective group homomorphism π1(''M'') → Hol(∇)/Hol0(∇) where π1(''M'') is the fundamental group of ''M'' which sends the homotopy class [γ] to the coset Pγ·Hol0(∇). *If ''M'' is simply connected then Hol(∇) = Hol0(∇). *∇ is flat (i.e. has vanishing curvature) iff Hol0(∇) is trivial. ===Riemannian holonomy groups=== The holonomy of a Riemannian manifold (''M'', ''g'') is the just holonomy group of the Levi-Civita connection on the tangent bundle to ''M''. A 'generic' ''n''-dimensional Riemannian manifold has an orthogonal group holonomy, or special orthogonal group if it is orientable manifold. Manifolds whose holonomy groups are proper subgroups of O(''n'') or SO(''n'') have special properties. In 1955, M. Berger gave a complete classification of possible holonomy groups for simply connected, Riemannian manifolds which are irreducible (not locally a product space) and nonsymmetric (not locally a Riemannian symmetric space). Berger's list is as follows: {| cellpadding=2 style="margin: auto; text-align: left; border-collapse: collapse;" border=1 ! Hol(''g'') || dim(''M'') || Type of manifold || Comments |- | Special orthogonal group || ''n'' || generic |- | Unitary group || 2''n'' || Kähler manifold || Kähler |- | Special unitary group || 2''n'' || Calabi-Yau manifold || Ricci-flat, Kähler |- |Sp(''n'')·Sp(1) || 4''n'' || quaternionic Kähler manifold || Einstein manifold |- | Symplectic group || 4''n'' || hyperkähler manifold || Ricci-flat, Kähler |- | G2 (mathematics) || 7 || G2 manifold || Ricci-flat |- | Spin group || 8 || Spin(7) manifold || Ricci-flat |} It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. Riemannian symmetric spaces, which are locally isometric to homogeneous spaces have local holonomy isomorphic to . These too have been completely classified. ===Special holonomy manifolds in string theory=== Riemannian manifolds with special holonomy play an important role in string theory compactification (physics). This is because special holonomy manifolds admit covariantly constant (parallel) spinors and thus preserve some fraction of the original supersymmetry. Most important are compactifications on Calabi-Yau manifolds with SU(2) or SU(3) holonomy. Also important are compactifications on G2 manifolds. ==On principal bundles== The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let ''P'' be a principal bundle over a smooth manifold ''M'' for some Lie group ''G'' and let ω be a connection form on ''P''. Given a piecewise smooth loop (topology) γ : [0,1] → ''M'' based at ''x'' in ''M'' and a point ''p'' in the fiber over ''x'' the connection defines a unique ''horizontal lift'' such that . The end point of the horizontal lift, , will not generally be ''p'' but rather some other point ''p''·''g'' in the fiber over ''x''. Define an equivalence relation ~ on ''P'' by saying that ''p''~''q'' if they can be joined by a piecewise smooth horizontal path in ''P''. The holonomy group of ω based at ''p'' is then defined as : The local holonomy group based at ''p'' is the subgroup coming from horizontal lifts of contractible loops γ. If ''M'' and ''P'' are connected space then the holonomy group depends on the basepoint ''p'' only up to conjugation in ''G''. Explicitly, : Moreover if ''p''~''q'' the Hol''p''(ω) = Hol''q''(ω). It is therefore customary to drop reference to the basepoint with the understanding that the definition is good up to conjugation. Some important properties of holonomy group include: *Hol0(ω) is a connected, Lie subgroup of ''G''. *Hol0(ω) is the identity component of Hol(ω). *There is a natural, surjective group homomorphism π1(''M'') → Hol(ω)/Hol0(ω). *If ''M'' is simply connected then Hol(ω) = Hol0(ω). *ω is flat (i.e. has vanishing curvature) iff Hol0(ω) is trivial. == References and external links == * Chi, Merkulov, and Schwachhöfer, On the incompleteness of Berger's list, [http://arxiv.org/abs/dg-ga/9508014 arXiv:dg-ga/9508014]. * Joyce, ''Compact Manifolds with Special Holonomy'', Oxford University Press, 2000. ISBN 0-19-850601-5. Differential geometry Riemannian geometry
The corrections by User:Serenus suggest a couple of things to me (I'm not an expert in these matters): (1) Is there a necessary distinction to be made, between holonomy and ''local'' holonomy? (2) Is this topic connected with the Berger list, which is a Requested Article? User:Charles Matthews 10:22, 10 May 2004 (UTC) Answering (2) myself, it seems clear from some Googling that it's 'yes' (for example http://arxiv.org/abs/dg-ga/9508014). But that recent work has shown up some gaps. So, redirecting Berger list here, and adding a note. User:Charles Matthews 10:30, 10 May 2004 (UTC)
See other meanings of words starting from letter:
Words begining with Holonomy:
Holonomy
Holonomy
Holonomy_group
Holonomy
In differential geometry, the holonomy group of a connection (mathematics) on a vector bundle over a smooth manifold ''M'' is the group (mathematics) of linear transformations induced by parallel transport around closed loops in ''M''. There is an analogous notion for connection form on principal bundles over ''M''. The holonomy group of a connection is intimately associated with the curvature form of that connection. The holonomy group of a Riemannian manifold ''M'' is the just holonomy group of the Levi-Civita connection on the tangent bundle of ''M''. ==On vector bundles== Let ''E'' be a rank ''k'' vector bundle over a smooth manifold ''M'' and let ∇ be a connection (mathematics) on ''E''. Given a piecewise smooth loop (topology) γ : [0,1] → ''M'' based at ''x'' in ''M'', the connection defines a parallel transport map . This map is both linear and invertible and so defines an element of GL(''Ex''). The holonomy group of ∇ based at ''x'' is defined as : The local holonomy group based at ''x'' is the subgroup coming from contractible loops γ. If ''M'' is connected space then the holonomy group depends on the basepoint ''x'' only up to conjugation in GL(''k'', R). Explicitly, if γ is a path from ''x'' to ''y'' in ''M'' then : Choosing different identifications of ''Ex'' with R''k'' also gives conjugate subgroups. It is therefore customary to drop reference to the basepoint with the understanding that the definition is good up to conjugation. Some important properties of holonomy group include: *Hol0(∇) is a connected, Lie subgroup of GL(''k'', R). *Hol0(∇) is the identity component of Hol(∇). *There is a natural, surjective group homomorphism π1(''M'') → Hol(∇)/Hol0(∇) where π1(''M'') is the fundamental group of ''M'' which sends the homotopy class [γ] to the coset Pγ·Hol0(∇). *If ''M'' is simply connected then Hol(∇) = Hol0(∇). *∇ is flat (i.e. has vanishing curvature) iff Hol0(∇) is trivial. ===Riemannian holonomy groups=== The holonomy of a Riemannian manifold (''M'', ''g'') is the just holonomy group of the Levi-Civita connection on the tangent bundle to ''M''. A 'generic' ''n''-dimensional Riemannian manifold has an orthogonal group holonomy, or special orthogonal group if it is orientable manifold. Manifolds whose holonomy groups are proper subgroups of O(''n'') or SO(''n'') have special properties. In 1955, M. Berger gave a complete classification of possible holonomy groups for simply connected, Riemannian manifolds which are irreducible (not locally a product space) and nonsymmetric (not locally a Riemannian symmetric space). Berger's list is as follows: {| cellpadding=2 style="margin: auto; text-align: left; border-collapse: collapse;" border=1 ! Hol(''g'') || dim(''M'') || Type of manifold || Comments |- | Special orthogonal group || ''n'' || generic |- | Unitary group || 2''n'' || Kähler manifold || Kähler |- | Special unitary group || 2''n'' || Calabi-Yau manifold || Ricci-flat, Kähler |- |Sp(''n'')·Sp(1) || 4''n'' || quaternionic Kähler manifold || Einstein manifold |- | Symplectic group || 4''n'' || hyperkähler manifold || Ricci-flat, Kähler |- | G2 (mathematics) || 7 || G2 manifold || Ricci-flat |- | Spin group || 8 || Spin(7) manifold || Ricci-flat |} It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds. The last two exceptional cases were the most difficult to find. Riemannian symmetric spaces, which are locally isometric to homogeneous spaces have local holonomy isomorphic to . These too have been completely classified. ===Special holonomy manifolds in string theory=== Riemannian manifolds with special holonomy play an important role in string theory compactification (physics). This is because special holonomy manifolds admit covariantly constant (parallel) spinors and thus preserve some fraction of the original supersymmetry. Most important are compactifications on Calabi-Yau manifolds with SU(2) or SU(3) holonomy. Also important are compactifications on G2 manifolds. ==On principal bundles== The definition for holonomy of connections on principal bundles proceeds in parallel fashion. Let ''P'' be a principal bundle over a smooth manifold ''M'' for some Lie group ''G'' and let ω be a connection form on ''P''. Given a piecewise smooth loop (topology) γ : [0,1] → ''M'' based at ''x'' in ''M'' and a point ''p'' in the fiber over ''x'' the connection defines a unique ''horizontal lift'' such that . The end point of the horizontal lift, , will not generally be ''p'' but rather some other point ''p''·''g'' in the fiber over ''x''. Define an equivalence relation ~ on ''P'' by saying that ''p''~''q'' if they can be joined by a piecewise smooth horizontal path in ''P''. The holonomy group of ω based at ''p'' is then defined as : The local holonomy group based at ''p'' is the subgroup coming from horizontal lifts of contractible loops γ. If ''M'' and ''P'' are connected space then the holonomy group depends on the basepoint ''p'' only up to conjugation in ''G''. Explicitly, : Moreover if ''p''~''q'' the Hol''p''(ω) = Hol''q''(ω). It is therefore customary to drop reference to the basepoint with the understanding that the definition is good up to conjugation. Some important properties of holonomy group include: *Hol0(ω) is a connected, Lie subgroup of ''G''. *Hol0(ω) is the identity component of Hol(ω). *There is a natural, surjective group homomorphism π1(''M'') → Hol(ω)/Hol0(ω). *If ''M'' is simply connected then Hol(ω) = Hol0(ω). *ω is flat (i.e. has vanishing curvature) iff Hol0(ω) is trivial. == References and external links == * Chi, Merkulov, and Schwachhöfer, On the incompleteness of Berger's list, [http://arxiv.org/abs/dg-ga/9508014 arXiv:dg-ga/9508014]. * Joyce, ''Compact Manifolds with Special Holonomy'', Oxford University Press, 2000. ISBN 0-19-850601-5. Differential geometry Riemannian geometry
Holonomy
The corrections by User:Serenus suggest a couple of things to me (I'm not an expert in these matters): (1) Is there a necessary distinction to be made, between holonomy and ''local'' holonomy? (2) Is this topic connected with the Berger list, which is a Requested Article? User:Charles Matthews 10:22, 10 May 2004 (UTC) Answering (2) myself, it seems clear from some Googling that it's 'yes' (for example http://arxiv.org/abs/dg-ga/9508014). But that recent work has shown up some gaps. So, redirecting Berger list here, and adding a note. User:Charles Matthews 10:30, 10 May 2004 (UTC)
See other meanings of words starting from letter:
H
HA | HB | HC | HD | HE | HF | HG | HI | HJ | HK | HL | HM | HN | HO | HP | HR | HS | HT | HU | HW | HX | HY | HZ |Words begining with Holonomy:
Holonomy
Holonomy
Holonomy_group
These materials are based on Wikipedia and licensed under the GNU FDL
YouTube.com videos better site than Turbo Tax 2007
