Restricted Holonomy Group Research Articles

Parallel Translations for a Left Invariant Spray

In this paper, we study the left invariant spray geometry on a connected Lie group. Using the technique of invariant frames, we find the ordinary differential equations on the Lie algebra describing for a left invariant spray structure the linearly parallel translations along a geodesic and the nonlinearly parallel translations along a smooth curve. In these equations, the connection operator plays an important role. Using parallel translations, we provide alternative interpretations or proofs for some homogeneous curvature formulae. In particular, the Riemannian curvture appears in both a double Lie derivative along the spray vector and brackets between smooth vector fields induced by the connection operator. We propose two questions in left invariant spray geometry. One question generalizes Landsberg Problem in Finsler geometry, and the other concerns the restricted holonomy group.

Lévy Laplacians, Holonomy Group and Instantons on 4-Manifolds

A connection between the Yang–Mills gauge fields on 4-dimensional orientable compact Riemannian manifolds and modified Lévy Laplacians is studied. A modified Lévy Laplacian is obtained from the Lévy Laplacian by the action of an infinite dimensional rotation. Under the assumption that the 4-manifold has a nontrivial restricted holonomy group of the bundle of self-dual 2-forms, the following is proved. There is a modified Lévy Laplacian such that a parallel transport in some vector bundle over the 4-manifold is a solution of the Laplace equation for this modified Lévy Laplacian if and only if the connection corresponding to the parallel transport satisfies the Yang–Mills anti-self-duality equations. An analogous connection between the Laplace equation for the Lévy Laplacian and the Yang–Mills equations was previously known.

On the holonomy of the Coulomb connection over manifolds with boundary

Narasimhan and Ramadas [Commun. Math. Phys. 67, 121 (1979)] showed that the restricted holonomy group of the Coulomb connection is dense in the connected component of the identity of the gauge group when one considers the product principal bundle S3×SU(2)→S3. Instead of a base manifold S3, we consider here a base manifold of dimension n≥2 with a boundary and use Dirichlet boundary conditions on connections as defined by Marini [Commun. Pure Appl. Math. 45, 1015 (1992)]. A key step in the method of Narasimhan and Ramadas consisted in showing that the linear space spanned by the curvature form at one specially chosen connection is dense in the holonomy Lie algebra with respect to an appropriate Sobolev norm. Our objective is to explore the effect of the presence of a boundary on this construction of the holonomy Lie algebra. Fixing appropriate Sobolev norms, it will be shown that the space spanned, linearly, by the curvature form at any one connection is never dense in the holonomy Lie algebra. In contrast, the linear space spanned by the curvature form and its first commutators at the flat connection is dense and, in the C∞ category, is in fact the entire holonomy Lie algebra. The former, negative, theorem is proven for a general principle bundle over M, while the latter, positive, theorem is proven only for a product bundle over the closure of a bounded open subset of Rn. Our technique for proving the absence of density consists in showing that the linear space spanned by the curvature form at one point is contained in the kernel of a linear map consisting of a third order differential operator, followed by a restriction operation at the boundary; this mapping is determined by the mean curvature of the boundary.

MINIMAL IMMERSIONS OF KÄHLER MANIFOLDS INTO EUCLIDEAN SPACES

It is proved here that a minimal isometric immersion of a Kähler-Einstein or homogeneous Kähler-manifold into an Euclidean space must be totally geodesic. As an application, it is shown that an open subset of the real hyperbolic plane RH2 cannot be minimally immersed into the Euclidean space. As another application, a proof is given that if an irreducible Kähler manifold is minimally immersed in a Euclidean space, then its restricted holonomy group must be U(n), where n = dimCM. 2000 Mathematics Subject Classification 53B25 (primary); 53C42 (secondary).

Recurrent tensors and holonomy group

Let M be a connected C∞. A method is being introduced here to study the action of the holonomy group and the restricted holonomy group of Γ on a recurrent tensor. The main result of this paper is that if the recurrence covector W of a recurrent tensor S on M is an exact form then the tensor S is invariant under the holonomy group of Γ and if W is a closed form then S is invariant under the restricted holonomy group of Γ. In the last section, this result is applied to some particular cases including the case of a riemannian manifold with recurrent curvature.

Some theorems on holonomy groups of Riemannian manifolds

For Riemannian manifolds there are four kinds of holonomy groups: the (nonrestricted) holonomy group H, the restricted holonomy group HO, the (nonrestricted) homogeneous holonomy group h, and the restricted homogeneous holonomy group h?. It is known that all of these are Lie groups of transformations and HO and hf are the connected components of the identity of H and h respectively. 1. Relations among invanant linear subspaces of h? and h. If the restricted homogeneous holonomy group h? is reducible (in the real number field) it is completely reducible, for h is a subgroup of the orthogonal group. If the holonomy group hf is reducible, we can take a repere in the tangent space ER(O) at the base point 0 of the holonomy group so that all elements of the group h? can be represented by matrices of the following type: