We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold 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 the complex Euclidean space 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 projective case) and basic facts of complex submanifolds.
The holonomy group of an (n+2)-dimensional simply-connected, indecomposable but non-irreducible Lorentzian manifold (M,h) is contained in the parabolic group $(\mathbb{R} \times SO(n))\ltimes \mathbb{R}^n$. The main ingredient of such a holonomy group is the SO(n)--projection $G:=pr_{SO(n)}(Hol_p(M,h))$ and one may ask whether it has to be a Riemannian holonomy group. In this paper we show that this is always the case, completing our results of the first part math.DG/0305139. We draw consequences for the existence of parallel spinors on Lorentzian manifolds.
The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_τ,\, 0<τ\leq 1$, are studied. Unlike the case of the standard sphere ($τ=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $τ^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_τ$ when $1 / (d+1) < τ^2 \leq 1$ and the stable ones are classified for $τ^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_τ$ with index …
We study associative submanifolds of the Berger space SO(5)/SO(3) endowed with its homogeneous nearly-parallel G2-structure. We focus on two geometrically interesting classes: the ruled associatives, and the associatives with special Gauss map. We show that the associative submanifolds ruled by a certain special type of geodesic are in correspondence with pseudo-holomorphic curves in $Gr_2^+(TS^4).$ Using this correspondence, together with a theorem of Bryant on superminimal surfaces in $S^4,$ we prove the existence of infinitely many topological types of compact immersed associative 3-folds i…
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 complete the local classification of normal holonomies for complex submanifolds. 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 sub…
In 1955, Berger \cite{Ber} gave a list of irreducible reductive representations which can occur as the holonomy of a torsion-free affine connection. This list was stated to be complete up to possibly a finite number of missing entries. In this paper, we show that there is, in fact, an infinite family of representations which are missing from this list, thereby showing the incompleteness of Berger's classification. Moreover, we develop a method to construct torsion-free connections with prescribed holonomy, and use it to give a complete description of the torsion-free affine connections with th…
In this note we prove that QR-submanifolds of the hyper-Kahler manifolds under some conditions admit the $G_2$ holonomy. We give simplest examples of such QR-submanifolds namely tori. We conjecture that all $G_2$ holonomy manifolds arise in this way.
In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichnérowicz property. In particular, we derive a classification of Lorentzian normal holonomy representations.
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 abou…
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 ${\rm U}(n)$.
The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and reduce it to a system of ODEs along a characteristic direction. We introduce a notion of higher dimensional holonomy map in analogy with the one-dimensional case [J. Differential Geom., 36(3):551-589, 1992], and we provide a characterization for singularities as well as a deforma…
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 [CD09, 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 hi…
We investigate parallel submanifolds of a Riemannian symmetric space $N$. The special case of a symmetric submanifold has been investigated by many authors before and is well understood. We observe that there is an intrinsic property of the second fundamental form which distinguishes full symmetric submanifolds from arbitrary full parallel submanifolds of $N$, usually called "1-fullness of $M$". Furthermore, for every parallel submanifold $M$ of $N$ we consider the pullback bundle $TN|M$ with its induced connection, which admits a distinguished parallel subbundle $osc M$, usually called the "s…
We prove a Berger-type theorem which asserts that if the orthogonal subgroup generated by the torsion tensor (pulled back to a point by parallel transport) of a metric connection with skew-symmetric torsion is not transitive on the sphere, then the space must be locally isometric to a Lie group with a bi-invariant metric or its symmetric dual (we assume the space to be locally irreducible). We also prove that a (simple) Lie group with a bi-invariant metric admits only two flat metric connections with skew-symmetric torsion: the two flat canonical connections. In particular, we get a refinement…
Let $π:P\to B$ be a smooth $G$-bundle over a compact Riemannian manifold $B$ and $c$ a smooth loop in $B$ of constant seed $a(>0)$, where $G$ is compact semi-simple Lie group. In this paper, we prove that the holonomy map ${\rm hol}_c:\mathcal A_P^{H^s}\to G$ is a homothetic submersion of coefficient $a$, where $s$ is a non-negative integer, $\mathcal A_P^{H^s}$ is the Hilbert space of all $H^s$-connections of the bundle $P$. In particular, we prove that, if $s=0$, then ${\rm hol}_c$ has minimal regularizable fibres. From this fact, we can derive that each component of the inverse image of any…
This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bun…
We study the uniqueness of minimal submanifolds and the stability of the mean curvature flow in several well-known model spaces of manifolds of special holonomy. These include the Stenzel metric on the cotangent bundle of spheres, the Calabi metric on the cotangent bundle of complex projective spaces, and the Bryant--Salamon metrics on vector bundles over certain Einstein manifolds. In particular, we show that the zero sections, as calibrated submanifolds with respect to their respective ambient metrics, are unique among compact minimal submanifolds and are dynamically stable under the mean cu…
Weakly-irreducible not irreducible subalgebras of $\so(1,n+1)$ were classified by L. Berard Bergery and A. Ikemakhen. In the present paper a geometrical proof of this result is given. Transitively acting isometry groups of Lobachevskian spaces and transitively acting similarity transformation groups of Euclidean spaces are classified.
A theorem of Lawson and Simons states that the only stable minimal submanifolds in complex projective spaces are complex submanifolds. We generalize their result to the cases of quaternionic and octonionic projective spaces. Our approach gives a unified viewpoint towards conformal and projective geometries.
We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities continue to hold for conformal metrics, and moreover, extend naturally to minimal submanifolds. In addition, we obtain explicit estimates for Laplace eigenvalues of minimal submanifolds in terms of geometric quantities of the ambient space.
The main result of this paper is that a Lorentzian manifold is locally conformally equivalent to a manifold with recurrent lightlike vector field and totally isotropic Ricci tensor if and only if its conformal tractor holonomy admits a 2-dimensional totally isotropic invariant subspace. Furthermore, for semi-Riemannian manifolds of arbitrary signature we prove that the conformal holonomy algebra of a $C$-space is a Berger algebra. For Ricci-flat spaces we show how the conformal holonomy can be obtained by the holonomy of the ambient metric and get results for Riemannian manifolds and plane wav…
Let $M$ be a most singular orbit of the isotropy representation of a simple symmetric space. Let $(ν_i, Φ_i)$ be an irreducible factor of the normal holonomy representation $(ν_pM, Φ(p))$. We prove that there exists a basis of a section $Σ_i\subset ν_i$ of $Φ_i$ 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 cu…
We develop a holonomy reduction procedure for general Cartan geometries. We show that, given a reduction of holonomy, the underlying manifold naturally decomposes into a disjoint union of initial submanifolds. Each such submanifold corresponds to an orbit of the holonomy group on the modelling homogeneous space and carries a canonical induced Cartan geometry. The result can therefore be understood as a `curved orbit decomposition'. The theory is then applied to the study of several invariant overdetermined differential equations in projective, conformal and CR-geometry. This makes use of an eq…
The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified, thus completing the work on this subject begun by M. Berger in the 1950s. The methods employed include representation theory, the theory of hermitian symmetric spaces, twistor theory, and Poisson geometry. The latter theory is especially important for the construction and classification of those torsion-free connections whose holonomy falls into one of the so-called `exotic' cases, i.e., those that were not included in Berger's original li…
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>=2, be a full and irreducible homogeneous submanifold of the sphere $S^{N-1}\subset R^N$ and 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 isoparam…
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 certain condition, and derive the constancy of the principal curvatures of such submanifolds. As its result, we can derive that such submanifolds are isoparametric.
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on exceptional holonomy, in two parts. Part I introduces the exceptional holonomy groups, and explains constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T^7/G or T^8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T^7 or T^8. Ther…
Due to its length and several inaccuracies, this article is no longer suggested for reading. Moreover, in the meantime certain results presented herein could be improved by the author in a non-trivial fashion. Instead, the reader is referred to the following two articles: The extrinsic holonomy Lie algebra of a parallel submanifold (arXiv:0904.2611), Extrinsic homogeneity of parallel submanifolds (arXiv:0904.2636).
Let M^7 a manifold with holonomy in G_2, and Y^3 an associative submanifold with boundary in a coassociative submanifold. In [5], the authors proved that M_{X,Y}, the moduli space of its associative deformations with boundary in the fixed X, has finite virtual dimension. Using Bochner's technique, we give a vanishing theorem that forces M_{X,Y} to be locally smooth.
In this paper, we study Lagrangian submanifolds of the homogeneous nearly Kähler $6$-dimensional unit sphere $\mathbb{S}^6(1)$. As the main result, we derive a Simons' type integral inequality in terms of the second fundamental form for compact Lagrangian submanifolds of $\mathbb{S}^6(1)$. Moreover, we show that the equality sign occurs if and only if the Lagrangian submanifold is either the totally geodesic $\mathbb{S}^3(1)$ or the Dillen-Verstraelen-Vrancken's Berger sphere $S^3$ discribed in J Math Soc Japan, 42: 565-584, 1990.
We classify the possible local holonomy groups of Weyl connections. The Berger-Simons theorem and the Merkulov-Schwachhöfer classification of holonomy groups of irreducible torsion-free connections leaves us with the remaining case, where the Weyl connection $D$ is reducible and non-closed. In this case, it was shown by F. Belgun and A. Moroianu that the Weyl structure is an adapted Weyl structure of a non-closed conformal product. Furthermore we prove that non-closed Einstein-Weyl product structures only exist in dimension $4$.
An explicit expression of the canonical 8-form on a Riemannian manifold with a Spin(9)-structure, in terms of the nine local symmetric involutions involved, is given. The list of explicit expressions of all the canonical forms related to Berger's list of holonomy groups is thus completed. Moreover, some results on Spin(9)-structures as G-structures defined by a tensor and on the curvature tensor of the Cayley planes, are obtained.
Bryant \cite{Br} proved the existence of torsion free connections with exotic holonomy, i.e. with holonomy that does not occur on the classical list of Berger \cite{Ber}. These connections occur on moduli spaces $\Y$ of rational contact curves in a contact threefold $\W$. Therefore, they are naturally contained in the moduli space $\Z$ of all rational curves in $\W$. We construct a connection on $\Z$ whose restriction to $\Y$ is torsion free. However, the connection on $\Z$ has torsion unless both $\Y$ and $\Z$ are flat. We also show the existence of a new exotic holonomy which is a certain si…
We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G$_2$ and Spin(7) manifolds, and describe fundamental results and techniques in the field.
On a closed connected oriented manifold $M$ we study the space $\mathcal{M}_\|(M)$ of all Riemannian metrics which admit a non-zero parallel spinor on the universal covering. Such metrics are Ricci-flat, and all known Ricci-flat metrics are of this form. We show the following: The space $\mathcal{M}_\|(M)$ is a smooth submanifold of the space of all metrics, and its premoduli space is a smooth finite-dimensional manifold. The holonomy group is locally constant on $\mathcal{M}_\|(M)$. If $M$ is spin, then the dimension of the space of parallel spinors is a locally constant function on $\mathcal…
We identify a strong stability condition on minimal submanifolds that implies uniqueness and dynamical stability properties. In particular, we prove a uniqueness theorem and a C^1 dynamical stability theorem of the mean curvature flow for minimal submanifolds that satisfy this condition. The latter theorem states that the mean curvature flow of any other submanifold in a C^1 neighborhood of such a minimal submanifold exists for all time, and converges exponentially to the minimal one. This extends our previous uniqueness and stability theorem [arXiv:1605.03645] which applies only to calibrated…
For the ``Hopf bundle'' $S^1\to S^{2n,1} \to {\mathbb H}^n$, horizontal lifts of simple closed curves are studied. Let $γ$ be a piecewise smooth, simple closed curve on a complete totally geodesic surface $S$ in the base space. Then the holonomy displacement along $γ$ is given by $$ V(γ)=e^{λA(γ) i} $$ where $A(γ)$ is the area of the region on the surface $S$ surrounded by $γ$; $λ=1/2 $ or 0 depending on whether $S$ is a complex submanifold or not. We also carry out a similar investigation for the complex Heisenberg group ${\mathbb R} \to {\mathcal H}^{2n+1} \to {\mathbb C}^n$.
Let $K(X)$ denote the set of projective structures on a compact Riemann surface $X$ whose holonomy representations are discrete. We will show that each component of the interior of $K(X)$ is holomorphically equivalent to a complex submanifold of the product of Teichmüller spaces and the holonomy representation of every projective structure in the interior of $K(X)$ is a quasifuchsian group.
We consider parallel submanifolds $M$ of a Riemannian symmetric space $N$ and study the question whether $M$ is extrinsically homogeneous in $N$\,, i.e.\ whether there exists a subgroup of the isometry group of $N$ which acts transitively on $M$\,. First, given a "2-jet" $(W,b)$ at some point $p\in N$ (i.e. $W\subset T_pN$ is a linear space and $b:W\times W\to W^\bot$ is a symmetric bilinear form)\,, we derive necessary and sufficient conditions for the existence of a parallel submanifold with extrinsically homogeneous tangent holonomy bundle which passes through $p$ and whose 2-jet at $p$ is …
We study the control system of a Riemannian manifold $M$ of dimension $n$ rolling on the sphere $S^n$. The controllability of this system is described in terms of the holonomy of a vector bundle connection which, we prove, is isomorphic to the Riemannian holonomy group of the cone $C(M)$ of $M$. Using Berger's list, we reduce the possible holonomies to a few families. In particular, we focus on the cases where the holonomy is the unitary and the symplectic group. In the first case, using the rolling formalism, we construct explicitly a Sasakian structure on $M$; and in the second case, we cons…
The holonomy algebra $\g$ of an indecomposable Lorentzian (n+2)-dimensional manifold $M$ is a weakly-irreducible subalgebra of the Lorentzian algebra $\so_{1,n+1}$. L. Berard Bergery and A. Ikemakhen divided weakly-irreducible not irreducible subalgebras into 4 types and associated with each such subalgebra $\g$ a subalgebra $\h\subset \so_n$ of the orthogonal Lie algebra. We give a description of the spaces $\R(\g)$ of the curvature tensors for algebras of each type in terms of the space $¶(\h)$ of $\h$-valued 1-forms on $\Real^n$ that satisfy the Bianchi identity and reduce the classificatio…
In the present paper, we give a necessary and sufficient condition for a Riemannian manifold $(M,g)$ to have a reducible action of a hyperbolic analogue of the holonomy group. This condition amounts to a decomposition of $(M,g)$ as a warped product of a special form, in analogy to the classical de Rham decomposition theorem for Riemannian manifolds. As a consequence of these results and Berger's classification of holonomy groups, we obtain a simple necessary and sufficient condition for the complete controllability of the system of $(M,g)$ rolling against the hyperbolic space.
Let $M$ and $N$ be Riemannian symmetric spaces and $f:M\to N$ be a parallel isometric immersion. We additionally assume that there exist simply connected, irreducible Riemannian symmetric spaces $M_i$ with $\dim(M_i)\geq 2$ for $i=1,...,r$ such that $M\cong M_1\times...\times M_r$ . As a starting point, we describe how the intrinsic product structure of $M$ is reflected by a distinguished, fiberwise orthogonal direct sum decomposition of the corresponding first normal bundle. Then we consider the (second) osculating bundle $\osc f$, which is a $\nabla^N$-parallel vector subbundle of the pullba…
We give a new construction of compact Riemannian 7-manifolds with holonomy $G_2$. Let $M$ be a torsion-free $G_2$-manifold (which can have holonomy a proper subgroup of $G_2$) such that $M$ admits an involution $ι$ preserving the $G_2$-structure. Then $M/{\langle ι\rangle}$ is a $G_2$-orbifold, with singular set $L$ an associative submanifold of $M$, where the singularities are locally of the form $\mathbb R^3 \times (\mathbb R^4 / \{\pm 1\})$. We resolve this orbifold by gluing in a family of Eguchi-Hanson spaces, parametrized by a nonvanishing closed and coclosed $1$-form $λ$ on $L$. Much of…
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Simila…
A singular riemannian 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 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 pro…
