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).

@@ Line 9: / Line 9: @@
 * [[Restricted holonomy group]]
 * [[Holonomy group of Riemannian metric]]
+==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 <math>p</math>
+* Now, for each point <math>m</math>, the fibre at that point is the set of all bases at <math>m</math> that can arise from the basis at <math>p</math> by means of [[transport along a curve]] using the connection <math>\nabla</math>.
+* 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 the structure group can be reduced (for the same reasons).