Holonomy group: Difference between revisions

Latest revision as of 22:09, 24 July 2011

Definition

Let $M$ be a differential manifold, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} a vector bundle over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla} a connection for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle E} . For a point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \in M} the holonomy group at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} is the subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GL(E_p)} comprising those linear transformations that arise as the holonomy of a loop at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} .

If the differential manifold is path-connected, the holonomy groups at distinct points are conjugate as subgroups of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle GL(E_p)} so we can talk of the holonomy group.

Related notions

Facts

Reduction of structure group to holonomy group

Any path-connected differential manifold can be treated as a principal bundle with structure group being the holonomy group. The description is as follows:

Pick (arbitrarily) a basis at a particular point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p}
Now, for each point Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} , the fibre at that point is the set of all bases at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} that can arise from the basis at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p} by means of transport along a curve using the connection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla} .
In particular, any two bases at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} differ by the holonomy of a loop, which lies in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} . Thus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H} acts freely and transitively on the fibre at each point.

Moreover the holonomy group is the smallest group to which the structure group can be reduced (for the same reasons).

Revision as of 19:46, 18 May 2008 (view source) Vipul (talk \| contribs) m (4 revisions) ← Older edit		Latest revision as of 22:09, 24 July 2011 (view source) Vipul (talk \| contribs) (→‎Reduction of structure group to holonomy group)
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	* In particular, any two bases at <math>m</math> differ by the [[holonomy of a loop]], which lies in <math>H</math>. Thus <math>H</math> acts freely and transitively on the fibre at each point.		* In particular, any two bases at <math>m</math> differ by the [[holonomy of a loop]], which lies in <math>H</math>. Thus <math>H</math> acts freely and transitively on the fibre at each point.

	Moreover the holonomy group is the smallest group to which ~~thestructure~~ group can be reduced (for the same reasons).		Moreover the holonomy group is the smallest group to which the structure group can be reduced (for the same reasons).