The second illustration of the new functionality in the DifferentialGeometry package for Maple 17 also involves the exceptional Lie algebra , but now in the context of infinitesimal holonomy.
Let be an dimensional Riemannian manifold with metric .
Let be a fixed point.
Then the holonomy group Holof the Riemannian manifold is the group of linear transformations defined by the parallel transport of vectors around closed loops starting at . The Lie algebra of the holonomy group is called the infinitesimal holonomy . According to the AmbroseSinger Theorem, the infinitesimal holonomy can be computed in terms of the curvature tensor of the metric and the covariant derivatives of the curvature tensor. For many years it was an open problem in differential geometry as to whether or not there was a metric whose infinitesimal holonomy was the exceptional Lie algebra This question was answered in the affirmative by R. Bryant in 1987.
In this example, use the new command InfinitesimalHolonomy to verify an example of a metric with infinitesimal holonomy (due to S. Salamon).
First, define coordinates for a 7dimensional manifold :
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The computations for this example are much more easily performed in an anholonomic frame. Set:
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Calculate the structure equations for this coframe and initialize:
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The metric you will study is:
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This metric is Ricci flat.
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The infinitesimal holonomy is 14dimensional.
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The holonomy algebra is easier to display if you first use the command CanonicalBasis to transform the output to standard form and then make the change of variables u = t^3.
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You can now proceed as in Example 1 to show that this Lie algebra is also the exceptional Lie algebra .
Finally, you can use CovariantlyConstantTensors and the new command InvariantTensorsAtAPoint to show that the metric above admits a covariantly constant (parallel) 3form. Any covariantly constant 3form must be pointwise invariant with respect to the infinitesimal holonomy algebra. First find these invariant 3forms.
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In terms of the original coordinate this becomes:
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Every covariantly constant 3form must be a multiply of this form so use this as the ansatz which you pass to the command CovariantlyConstantTensors.
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