holonomy group वाक्य
उदाहरण वाक्य
मोबाइल
- The theory of isoparametric submanifolds is deeply related to the theory of holonomy groups.
- Lifts of loops about a point give rise to the holonomy group at that point.
- If F has a holonomy group then every leaf of F is compact with finite holonomy group ."
- If F has a holonomy group then every leaf of F is compact with finite holonomy group ."
- See below . ) It is now known that all of these possibilities occur as holonomy groups of Riemannian manifolds.
- It was not until much later that holonomy groups would be used to study Riemannian geometry in a more general setting.
- The paper " Submanifolds with constant principal curvatures and normal holonomy groups " is a very good introduction to such theory.
- As with the holonomy groups, the holonomy bundle also transforms equivariantly within the ambient principal bundle " P ".
- Theorem : " Let F be a holonomy group, then all the leaves of F are compact with finite holonomy group ."
- Theorem : " Let F be a holonomy group, then all the leaves of F are compact with finite holonomy group ."
- From these lists, an analogue of Simons's result that Riemannian holonomy groups act transitively on spheres may be observed : the complex holonomy representations are all prehomogeneous vector spaces.
- The parallel transport automorphisms defined by all closed curves based at " x " form a transformation group called the holonomy group of & nabla; at " x ".
- Conversely, there are examples of manifolds with these holonomy groups, such as the K3 surface, which are spin and have nonzero ?-invariant, hence are strongly scalar-flat.
- One of the earliest fundamental results on Riemannian holonomy is the theorem of, which asserts that the holonomy group is a closed Lie subgroup of O ( " n " ).
- The search for examples ultimately led to a complete classification of irreducible affine holonomies by Merkulov and Schwachh�fer ( 1999 ), with Bryant ( 2000 ) showing that every group on their list occurs as an affine holonomy group.
- The connection and curvature of any Riemannian manifold are closely related, the theory of holonomy groups, which are formed by taking linear maps defined by parallel transport around curves on the manifold, providing a description of this relationship.
- Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces ( that are locally isomorphic to the Cayley projective plane ), and the second does not occur at all as a holonomy group.
- Finally one checks that the first of these two extra cases only occurs as a holonomy group for locally symmetric spaces ( that are locally isomorphic to the Cayley projective plane ), and the second does not occur at all as a holonomy group.
- In 1952 Georges de Rham proved the " de Rham decomposition theorem ", a principle for splitting a Riemannian manifold into a Cartesian product of Riemannian manifolds by splitting the tangent bundle into irreducible spaces under the action of the local holonomy groups.
- On the way to his classification of Riemannian holonomy groups, Berger developed two criteria that must be satisfied by the Lie algebra of the holonomy group of a torsion-free affine connection which is not locally symmetric : one of them, known as " Berger's first criterion ", is a consequence of the Ambrose Singer theorem, that the curvature generates the holonomy algebra; the other, known as " Berger's second criterion ", comes from the requirement that the connection should not be locally symmetric.
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