Normal Holonomy Research Articles

A rigidity result for Kählerian manifolds endowed with closed conformal vector fields

We show that if a connected compact K¨ahlerian surface M with nonpositive Gaussian curvature is endowed with a closed conformal vector field ξ whose singular points are isolated, then M is isometric to a flat torus and ξ is parallel. We also consider the case of a connected complete K¨ahlerian manifod M of complex dimension n > 1 and endowed with a nontrivial closed conformal vector field ξ. In this case, it is well known that the singularities of ξ are automatically isolated and the nontrivial leaves of the distribution generated by ξ and Jξ are totally geodesic in M. Assuming that one such leaf is compact, has torsion normal holonomy group and that the holomorphic sectional curvature of M along it is nonpositive, we show that ξ is parallel and M is foliated by a family of totally geodesic isometric tori and also by a family of totally geodesic isometric complete K¨ahlerian manifolds of complex dimension n − 1. In particular, the universal covering of M is isometric to a Riemannian product having R2 as a factor. We also comment on a generic class of compact complex symmetric spaces not possessing nontrivial closed conformal vector fields, thus showing that we cannot get rid of the hypothesis of nonpositivity of the holomorphic sectional curvature in the direction of ξ.

Submanifolds, holonomy, and homogeneous geometry

This is an expository article. We would like to draw the attention to some problems in submanifold and homogeneous geometry related to the so-called normal holonomy. We will also survey on recent results obtained in cooperation with J. Berndt, A. J. Di Scala, S. Reggiani, J. S. Rodriguez, H. Tamaru, and F. Vittone.

Normal holonomy and rational properties of the shape operator

Let M be a most singular orbit of the isotropy representation of a simple symmetric space. Let $$(\nu _i, \Phi _i)$$ be an irreducible factor of the normal holonomy representation $$(\nu _pM, \Phi (p))$$ . We prove that there exists a basis of a section $$\Sigma _i\subset \nu _i$$ of $$\Phi _i$$ such that the corresponding shape operators have rational eigenvalues (this is not in general true for other isotropy orbits). Conversely, this property, if referred to some non-transitive irreducible normal holonomy factor, characterizes the isotropy orbits. We also prove that the definition of a submanifold with constant principal curvatures can be given by using only the traceless shape operator, instead of the shape operator, restricted to a non-transitive (non necessarily irreducible) normal holonomy factor. This article generalizes previous results of the authors that characterized Veronese submanifolds in terms of normal holonomy.

Cartan\u2019s incomplete classification and an explicit ambient metric of holonomy mathrm{G}_2^*

In his 1910 “Five Variables” paper, Cartan solved the equivalence problem for the geometry of (2, 3, 5) distributions and in doing so demonstrated an intimate link between this geometry and the exceptional simple Lie groups of type mathrm{G}_2. He claimed to produce a local classification of all such (complex) distributions which have infinitesimal symmetry algebra of dimension at least 6 (and which satisfy a natural uniformity condition), but in 2013 Doubrov and Govorov showed that this classification misses a particular distribution mathbf {E}. We produce a closed form for the Fefferman–Graham ambient metric {smash {{smash {widetilde{g}}}}}_{mathbf {E}} of the conformal class induced by (a real form of) mathbf {E}, expanding the small catalogue of known explicit, closed-form ambient metrics. We show that the holonomy group of smash {{smash {widetilde{g}}}_{mathbf {E}}} is the exceptional group {smash {mathrm{G}}_2^*} and use that metric to give explicitly a projective structure with normal projective holonomy equal to that group. We also present some simple but apparently novel observations about ambient metrics of general left-invariant conformal structures that were used in the determination of the explicit formula for smash {{smash {widetilde{g}}}_{mathbf {E}}}.

Mok's characteristic varieties and the normal holonomy group

In this paper we complete the study of the normal holonomy groups of complex submanifolds (non nec. complete) of Cn or CPn. We show that irreducible but non-transitive normal holonomies are exactly the Hermitian s-representations of [4, Table 1] (see Corollary 1.1). For each one of them we construct a non necessarily complete complex submanifold whose normal holonomy is the prescribed s-representation.We also show that if the submanifold has irreducible non-transitive normal holonomy then it is an open subset of the smooth part of one of the characteristic varieties studied by N. Mok in his work about rigidity of locally symmetric spaces.Finally, we prove that if the action of the normal holonomy group of a projective submanifold is reducible then the submanifold is an open subset of the smooth part of a so called join, i.e. the union of the lines joining two projective submanifolds.

The normal holonomy of $CR$-submanifolds

We study the normal holonomy group, i.e. the holonomy group of the normal connection, of a $CR$-submanifold of a complex space form. We show that the normal holonomy group of a coisotropic submanifold acts as the holonomy representation of a Riemannian symmetric space. In case of a totally real submanifold we give two results about reduction of codimension. We describe explicitly the action of the normal holonomy in the case in which the totally real submanifold is contained in a totally real totally geodesic submanifold. In such a case we prove the compactness of the normal holonomy group.

Normal holonomy of orbits and Veronese submanifolds

It was conjectured, twenty years ago, the following result that would generalize the so-called rank rigidity theorem for homogeneous Euclidean submanifolds: let $M^n$, $n\geq2$, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset \mathbb{R}^N$ such that the normal holonomy group is not transitive (on the unit sphere of the normal space to the sphere). Then $M^n$ must be an orbit of an irreducible $s$-representation (i.e. the isotropy representation of a semisimple Riemannian symmetric space). If $n=2$, then the normal holonomy is always transitive, unless $M$ is a homogeneous isoparametric hypersurface of the sphere (and so the conjecture is true in this case). We prove the conjecture when $n=3$. In this case $M^3$ must be either isoparametric or a Veronese submanifold. The proof combines geometric arguments with (delicate) topological arguments that use information from two different fibrations with the same total space (the holonomy tube and the caustic fibrations). We also prove the conjecture for $n\geq 3$ when the normal holonomy acts irreducibly and the codimension is the maximal possible $n(n+1)/2$. This gives a characterization of Veronese submanifolds in terms of normal holonomy. We also extend this last result by replacing the homogeneity assumption by the assumption of minimality (in the sphere). Another result of the paper, used for the case $n=3$, is that the number of irreducible factors of the local normal holonomy group, for any Euclidean submanifold $M^n$, is less or equal than $[n/2]$ (which is the rank of the orthogonal group $SO(n)$). This bound is sharp and improves the known bound $n(n-1)/2$.

The constancy of principal curvatures of curvature-adapted submanifolds in symmetric spaces

In this paper, we investigate complete curvature-adapted submanifolds with maximal flat section and trivial normal holonomy group in symmetric spaces of compact type or non-compact type under a certain condition, and derive the constancy of the principal curvatures of such submanifolds. As a result, we derive that such submanifolds are isoparametric.

Surfaces in $${\mathbb{R}^4}$$ with constant principal angles with respect to a plane

We study surfaces in \({\mathbb{R}^4}\) whose tangent spaces have constant principal angles with respect to a plane. Using a PDE we prove the existence of surfaces with arbitrary constant principal angles. The existence of such surfaces turns out to be equivalent to the existence of a special local symplectomorphism of \({\mathbb{R}^2}\). We classify all surfaces with one principal angle equal to 0 and observe that they can be constructed as the union of normal holonomy tubes. We also classify the complete constant angles surfaces in \({\mathbb{R}^4}\) with respect to a plane. They turn out to be extrinsic products. We characterize which surfaces with constant principal angles are compositions in the sense of Dajczer-Do Carmo. Finally, we classify surfaces with constant principal angles contained in a sphere and those with parallel mean curvature vector field.

A Berger type normal holonomy theorem for complex submanifolds

We prove a Berger type theorem for the normal holonomy \({\Phi^\perp}\) (i.e., the holonomy group of the normal connection) of a full complete complex submanifold M of the complex projective space \({\mathbb{C} P^n}\). Namely, if \({\Phi^\perp}\) does not act transitively, then M is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of \({\mathbb{C}^n}\) the normal holonomy is generic, i.e., it acts transitively on the unit sphere of the normal space. The methods in the proofs rely heavily on the singular data of appropriate holonomy tubes (after lifting the submanifold to the complex Euclidean space, in the \({\mathbb{C} P^n}\) case) and basic facts of complex submanifolds.

Maximal totally complex submanifolds of ℍℙ n : homogeneity and normal holonomy

We prove that a maximal totally complex submanifold $N^{2n}$ of the quaternionic projective space $\mathbb{H}\mathbb{P}^n$ ($n\geq 2$) is a parallel submanifold, provided one of the following conditions is satisfied: (1) $N$ is the orbit of a compact Lie group of isometries, (2) the restricted normal holonomy is a proper subgroup of ${\rm U}(n)$.

NORMAL HOLONOMY AND WRITHING NUMBER OF SMOOTH KNOTS

The normal holonomy of a smooth knot in Euclidean space is a geometric invariant with value in the unit circle which is closely related to the writhing number. Here we prove that normal holonomy fibers the space of smooth knots, except for possible singularities on the set of round knots. We deduce results on the existence of isotopies of constant holonomy, writhe and twist.

Parallel submanifolds of complex projective space and their normal holonomy

The object of this article is to compute the holonomy group of the normal connection of complex parallel submanifolds of the complex projective space. We also give a new proof of the classification of complex parallel submanifolds by using a normal holonomy approach. Indeed, we explain how these submanifolds can be regarded as the unique complex orbits of the (projectivized) isotropy representation of an irreducible Hermitian symmetric space. Moreover, we show how these important submanifolds are related to other areas of mathematics and theoretical physics. Finally, we state a conjecture about the normal holonomy group of a complete and full complex submanifold of the complex projective space.

Nonnegatively curved vector bundles with large normal holonomy groups

When B B is a biquotient, we show that there exist vector bundles over B B with metrics of nonnegative curvature whose normal holonomy groups have arbitrarily large dimension.

Singular Riemannian foliations on simply connected spaces

A singular foliation on a complete Riemannian manifold is said to be Riemannian if each geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. The singular foliation is said to admit sections if each regular point is contained in a totally geodesic complete immersed submanifold that meets every leaf orthogonally and whose dimension is the codimension of the regular leaves. A typical example of such a singular foliation is the partition by orbits of a polar action, e.g. the orbits of the adjoint action of a compact Lie group on itself. We prove that a singular Riemannian foliation with compact leaves that admits sections on a simply connected space has no exceptional leaves, i.e., each regular leaf has trivial normal holonomy. We also prove that there exists a convex fundamental domain in each section of the foliation and in particular that the space of leaves is a convex Coxeter orbifold.

A geometric proof of the Berger Holonomy Theorem

We give a geometric proof of the Berger Holonomy Theorem. The proof uses Euclidean submanifold geometry of orbits and gives a link between Riemannian holonomy groups and normal holonomy groups.

The Normal Holonomy Group of Kähler Submanifolds

We study the (restricted) holonomy group Hol(∇⊥) of the normal connection ∇⊥ (shortened to normal holonomy group) of a Kähler submanifold of a complex space form. We prove that if the normal holonomy group acts irreducibly on the normal space then it is linear isomorphic to the holonomy group of an irreducible Hermitian symmetric space. In particular, it is a compact group and the complex structure J belongs to its Lie algebra.We prove that the normal holonomy group acts irreducibly if the submanifold is full (that is, it is not contained in a totally geodesic proper Kähler submanifold) and the second fundamental form at some point has no kernel. For example, a Kähler–Einstein submanifold of CPn has this property.We define a new invariant μ of a Kähler submanifold of a complex space form. For non‐full submanifolds, the invariant μ measures the deviation of J from belonging to the normal holonomy algebra. For a Kähler–Einstein submanifold, the invariant μ is a rational function of the Einstein constant. By using the invariant μ, we prove that the normal holonomy group of a not necessarily full Kähler–Einstein submanifold of CPn is compact, and we give a list of possible holonomy groups.The approach is based on a definition of the holonomy algebra hol(P) of an arbitrary curvature tensor field P on a vector bundle with a connection and on a De Rham type decomposition theorem for hol(P). 2000 Mathematics Subject Classification 53C40 (primary), 53B25 (secondary).

The push-out region of an immersed manifold

We consider immersions: $ f:M^{m} \rightarrow {\hbox\mathbb{R}}^n $ and construct a subspace $ \Omega (f) $ of $ {\hbox{\mathbb{R}}}^{n-m} $ which corresponds to a set of embedded manifolds which are either parallel to f, tubes around f or, in general, partial tubes around f. This space is invariant under the action of the normal holonomy group, $ \mathcal{H} ol (f) $ We investigate the case where $ \mathcal{H} ol (f) $ is non-trivial and obtain some results on the number of connected components of $ \Omega (f) $ .

Normal holonomy and writhing number of polygonal knots

The normal holonomy of a polygonal knot is a geometrical invariant which is closely related to the writhing number. We show that normal holonomy fibers the space of knots over the circle and deduce that the writhing number fibers the space of knots over the real line. Consequently, two isotopic knots which have the same writhing number are isotopic through a family of knots having the same writhing number. In a similar vein, two isotopic knots having zero holonomy are isotopic through a family of such knots if and only if they have the same autoparallel linking number. More generally, the definition of normal holonomy makes sense for immersed polygonal knots. This time normal holonomy fibers the space of immersed knots over the circle, but now there are only two isotopy classes of immersed knots of zero holonomy.

Volume growth and holonomy in nonnegative curvature

The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between the codimension of the soul and the maximal dimension of an orbit of the action of the normal holonomy group of the soul. Additionally, an example of a simply-connected soul with a non-compact normal holonomy group is constructed.