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.
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…
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…
In this paper, we consider projective deformation of the geodesic system of Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$ and a holonomy invariant function $P$, we investigate the metrizability property of the projective deformation $\widetilde{S}=S-2λP C$. We prove that for any holonomy invariant nontrivial function $P$ and for almost every value $λ\in R$, such deformation is not Finsler metrizable. We identify the cases where such deformation can lead to a metrizable spray: in these cases, the holonomy invariant function is necessarily one of the principal c…
In this paper, we investigate the holonomy structure of the most accessible and demonstrative 2-dimensional Finsler surfaces, the Randers surfaces. Randers metrics can be considered as the solutions of the Zermelo navigation problem. We give the classification of the holonomy groups of locally projectively flat Randers two-manifolds of constant curvature. In particular, we prove that the holonomy group of a simply connected non-Riemannian projectively flat Finsler two-manifold of constant non-zero flag curvature is maximal and isomorphic to the orientation preserving diffeomorphism group of th…
For a given quasi-Fuchsian representation $ρ:π_1(S)\to$ PSL$_2\mathbb{C}$ of the fundamental group of a closed surface $S$ of genus $g\geq 2$, we prove that a generic branched complex projective structure on $S$ with holonomy $ρ$ and two branch points is obtained by bubbling some unbranched structure on $S$ with the same holonomy.
The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type $(1,1)$ curvature is a submanifold of a submaximally symmetric quaternionic model, and show how this fits into the construction. We give conditions for when …
Minimal Submanifolds of The Complex and Quaternionic Projective and Hyperbolic Spaces $\cn P^{2n-1}$, $\hn P^{2n-1}$, $\cn H^{2n-1}$, $\hn H^{2n-1}$ via Harmonic Morphisms
We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurf…
We provide a classification theorem for compact stable minimal immersions (CSMI) of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective space with any other Riemannian manifold. We characterize the complex minimal immersions of codimension $2$ or dimension $2$ as the only CSMI in the product of two complex projective spaces. As an application, we characterize the CSMI of codimension $1$ or dimension $1$ (codimension $1$ and $2$ or dimension $1$ and $2$) in the product of a complex (quaternionic) projective sp…
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.
We prove that there do not exist odd-dimensional stable compact minimal immersions in the product of two complex projective spaces. We also prove that the only stable compact minimal immersions in the product of a quaternionic projective space with any other Riemannian manifold are the products of quaternionic projective subspaces with compact stable minimal immersions of the second manifold in the Riemmanian product. These generalize similar results of the second-named author of immersions with low dimensions or codimensions to immersions with arbitrary dimensions. In addition, we prove that …
We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate ap…
In this paper we investigate the properties of the real and complex projective structures associated to Hitchin and quasi-Hitchin representations that were originally constructed using Guichard-Wienhard's theory of domains of discontinuity. We determine the topology of the underlying manifolds and we prove that some of these geometric structures are fibered in a special standard way. In order to prove these results, we give two new ways to construct these geometric structures: we construct them using gauge theory, flat bundles and Higgs bundles, and we also give a new geometric way to construc…
