On the dimension of totally geodesic submanifolds in the Prym loci

In this paper we give a bound on the dimension of a totally geodesic submanifold of the moduli space of polarised abelian varieties of a given dimension, which is contained in the Prym locus of a (possibly) ramified double cover. This improves the already known bounds. The idea is to adapt the techniques introduced by the authors in collaboration with A. Ghigi and G. P. Pirola for the Torelli map to the case of the Prym maps of (ramified) double covers.

The index conjecture for symmetric spaces

In 1980, Oniščik [A. L. Oniščik, Totally geodesic submanifolds of symmetric spaces, Geometric methods in problems of algebra and analysis. Vol. 2, Yaroslav. Gos. Univ., Yaroslavl’ 1980, 64–85, 161] introduced the index of a Riemannian symmetric space as the minimal codimension of a (proper) totally geodesic submanifold. He calculated the index for symmetric spaces of rank {\leq 2}, but for higher rank it was unclear how to tackle the problem. In [J. Berndt, S. Console and C. E. Olmos, Submanifolds and holonomy, 2nd ed., Monogr. Res. Notes Math., CRC Press, Boca Raton 2016], [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], [J. Berndt and C. Olmos, The index of compact simple Lie groups, Bull. Lond. Math. Soc. 49 2017, 5, 903–907], [J. Berndt and C. Olmos, On the index of symmetric spaces, J. reine angew. Math. 737 2018, 33–48], [J. Berndt, C. Olmos and J. S. Rodríguez, The index of exceptional symmetric spaces, Rev. Mat. Iberoam., to appear] we developed several approaches to this problem, which allowed us to calculate the index for many symmetric spaces. Our systematic approach led to a conjecture, formulated first in [J. Berndt and C. Olmos, Maximal totally geodesic submanifolds and index of symmetric spaces, J. Differential Geom. 104 2016, 2, 187–217], for how to calculate the index. The purpose of this paper is to verify the conjecture.

Axisymmetric Diffeomorphisms and Ideal Fluids on Riemannian 3-Manifolds

We study the Riemannian geometry of 3D axisymmetric ideal fluids. We prove that the $L^2$ exponential map on the group of volume-preserving diffeomorphisms of a $3$-manifold is Fredholm along axisymmetric flows with sufficiently small swirl. Along the way, we define the notions of axisymmetric and swirl-free diffeomorphisms of any manifold with suitable symmetries and show that such diffeomorphisms form a totally geodesic submanifold of infinite $L^2$ diameter inside the space of volume-preserving diffeomorphisms whose diameter is known to be finite. As examples, we derive the axisymmetric Euler equations on $3$-manifolds equipped with each of Thurston’s eight model geometries.

On a property of W4-manifolds

The properties of almost Hermitian manifolds belonging to the Gray — Hervella class W4 are considered. The almost Hermitian manifolds of this class were studied by such outstanding geometers like Alfred Gray, Izu Vaisman, and Vadim Feodorovich Kirichenko. Using the Cartan structural equations of an almost contact metric structure induced on an arbitrary oriented hypersurface of a W4-manifold, some results on totally umbilical and totally geodesic hypersurfaces of W4-manifolds are presented. It is proved that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a W4-manifold is either homothetic to a Sasakian structure or cosymplectic. Moreover, the quasi-Sasakian structure is cosymplectic if and only if the hypersurface is a to­tally geodesic submanifold of the considered W4-manifold. From the present result it immediately follows that the quasi-Sasakian structure induced on a totally umbilical hypersurface of a locally confor­mal Kählerian (LCK-) manifold also is either homothetic to a Sasakian structure or cosymplectic.

Exponentially Harmonic Maps into Spheres

We study smooth exponentially harmonic maps from a compact, connected, orientable Riemannian manifold M into a sphere S m ⊂ R m + 1 . Given a codimension two totally geodesic submanifold Σ ⊂ S m , we show that every nonconstant exponentially harmonic map ϕ : M → S m either meets or links Σ . If H 1 ( M , Z ) = 0 then ϕ ( M ) ∩ Σ ≠ ∅ .

On splitting rank of non-compact-type symmetric spaces and bounded cohomology

Let [Formula: see text] be a higher rank symmetric space of non-compact type where [Formula: see text]. We define the splitting rank of [Formula: see text], denoted by [Formula: see text], to be the maximal dimension of a totally geodesic submanifold [Formula: see text] which splits off an isometric [Formula: see text]-factor. We compute explicitly the splitting rank for each irreducible symmetric space. For an arbitrary (not necessarily irreducible) symmetric space, we show that the comparison map [Formula: see text] is surjective in degrees [Formula: see text], provided [Formula: see text] has no direct factors of [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]. This generalizes the result of [J.-F. Lafont and S. Wang, Barycentric straightening and bounded cohomology, to appear in J. Eur. Math. Soc.] regarding Dupont’s problem.

On the index of symmetric spaces

LetMbe an irreducible Riemannian symmetric space. The index ofMis the minimal codimension of a (nontrivial) totally geodesic submanifold ofM. We prove that the index is bounded from below by the rank of the symmetric space. We also classify the irreducible Riemannian symmetric spaces whose index is less than or equal to 3.

On totally geodesic submanifolds in the Jacobian locus

We study submanifolds of Ag that are totally geodesic for the locally symmetric metric and which are contained in the closure of the Jacobian locus but not in its boundary. In the first section we recall a formula for the second fundamental form of the period map Mg ↪ Ag due to Pirola, Tortora and the first author. We show that this result can be stated quite neatly using a line bundle over the product of the curve with itself. We give an upper bound for the dimension of a germ of a totally geodesic submanifold passing through [C] ∈ Mg in terms of the gonality of C. This yields an upper bound for the dimension of a germ of a totally geodesic submanifold contained in the Jacobian locus, which only depends on the genus. We also study the submanifolds of Ag obtained from cyclic covers of ℙ1. These have been studied by various authors. Moonen determined which of them are Shimura varieties using deep results in positive characteristic. Using our methods we show that many of the submanifolds which are not Shimura varieties are not even totally geodesic.