Geometric Hydrodynamics on Flatland

Klas Modin (Chalmers University of Technology, Sweden)

Euler's equations for the motion of an incompressible perfect fluid possess a rich geometric structure, as revealed over the years by Poincaré, Arnold, and many others. The structure is particularly striking in two dimensions, where it gives rise to infinitely many conservation laws (Casimir functions). In this mini-course I will give an overview of geometric hydrodynamics, with an emphasis on the two-dimensional case and why its long-time dynamics (turbulence) is so different from the three- and higher-dimensional setting. I will also discuss how results from quantisation theory can be used to construct numerical discretizations that preserve the underlying geometry - an approach due to V. Zeitlin.

Noncommutative Geometry and Quantum Groups

Anna Pachol (University of South-Eastern Norway, Norway)

The main idea behind the noncommutative geometry is to “algebralize” geometric notions and then generalize them to noncommutative algebras. This way noncommutative geometry offers a generalised notion of the geometry. Quantum groups or Hopf algebras play the role of 'group objects' in noncommutative geometry and they provide an approach to the development of the theory much as Lie groups do in differential geometry. The term "quantum group" first appeared in the theory of quantum integrable systems and later was formalized by V. Drinfeld and M. Jimbo as a particular class of Hopf algebras with connection to deformation theory (as deformation of universal enveloping Lie algebra). Such deformations are classified in terms of classical r-matrices satisfying the classical Yang-Baxter equation.

We will start with the quantization of Poisson-Lie groups (i.e. Lie groups equipped with a Poisson structure) which provide a natural example of quantum groups. Other examples of Hopf algebras arising from Lie algebras through their universal enveloping algebras will also be discussed.

Local symplectic groupoids and applications

Alejandro Cabrera (Universidade Federal do Rio de Janeiro, Brazil)

In this talk, we shall review the concept of local symplectic groupoid and its relation to the Poisson geometry of its units space. We will present some recent results about such structures including a construction based on ODE's ("sprays"), an explicit description of the underlying symplectic realizations, and the construction of generating functions. Finally, we describe some applications to quantization of Poisson brackets and some (tentative) applications to the discretization of hamiltonian flows in Poisson manifolds.

Embedding tensors on Lie $\infty$-algebras with respect to Lie $\infty$-actions

Raquel Caseiro (University of Coimbra, Portugal)

Embedding tensors have been widely used and studied in the context of supergravity theory and gauge (higher) theories. We will look at embedded tensors from a purely mathematical perspective, starting by explaining the link between embedded tensors, Leibniz algebras and Lie $\infty$-algebras. Next, we will explore their homotopy versions: homotopy embedded tensors and non-abelian homotopy embedded tensors. These can be seen as Maurer-Cartan elements of a particular Lie $\infty$-algebra, they define a new Loday $\infty$-structure and are a morphism between Loday $\infty$-algebras.

From Poisson geometry to pseudogroups and geometric structures

Francesco Cattafi (Universität Würzburg, Germany)

The space of (local) symmetries of a given geometric structure has the natural structure of a Lie (pseudo)group. Conversely, geometric structures admitting a local model can be described via the pseudogroup of symmetries of such local model.

This philosophy can be made precise at various levels of generality (depending on the definition of "geometric structure") and using different tools/methods. In this talk I will present a new framework, which include previous formalisms (e.g. G-structures or Cartan geometries) and allows us to prove integrability theorems. The novelty of this point of view consists however in the fact that it uncovers the (beautiful!) hidden structures behind Lie pseudogroups and geometric structures. Indeed, the relevant objects which make this approach work are Lie groupoids endowed with a multiplicative "PDE-structure", their principal actions and the related Morita theory. Poisson geometry provides the guiding principle to understand those objects, which are directly inspired from, respectively, symplectic groupoids, principal Hamiltonian bundles and symplectic Morita equivalence.

This is based on joint work with Luca Accornero, Marius Crainic and María Amelia Salazar.

Stochastic variational principles for dissipative equations

Ana Bela Cruzeiro (Universidade de Lisboa, Portugal)

We derive deterministic dissipative equations of motion from variational principles for random Lagrangian paths. These are ordinary differential equations when the configuration space is finite dimensional and partial differential equations in the infinite dimensional case. To achieve this goal we introduce symmetry reduction for stochastic Lagrangian systems whose configuration space is a Lie group.

This is joint work with X. Chen and T. S. Ratiu.

Extending spheres to new examples of isoparametric hypersurfaces in symmetric spaces

Víctor Sanmartín López (Universidade de Santiago de Compostela, Spain)

A hypersurface is said to be isoparametric if it and its nearby equidistant hypersurfaces have constant mean curvature. In this talk, we will see examples of these objects in the context of symmetric spaces together with some well-known classification results. After that, we will focus on the new examples of isoparametric hypersurfaces with novel properties that we have constructed in symmetric space of non-compact type and rank greater than two. Finally, we will see how these examples can be somehow obtained from "extending" spheres in Euclidean spaces.

Classes of coadjoint orbits of the volume preserving diffeomorphism group

Cornelia Vizman (West University of Timisoara, Romania)

We describe classes of coadjoint orbits of the volume preserving diffeomorphism group that may accommodate singular vorticities for ideal fluids. Some coadjoint orbits arise via symplectic reduction in dual pairs: the Marsden-Weinstein ideal fluid dual pair and a new variant of the Holm-Marsden EPDiff dual pair.

Homogeneous Lorentzian critical metrics in dimension three

Sandro Caeiro Oliveira (Universidade de Vigo, Spain)

Critical metrics for the Hilbert-Einstein functional in dimension three are very rigid since they are of constant sectional curvature. Substituting the scalar curvature by a quadratic curvature invariant, one can consider the functionals \[ \begin{aligned} \mathcal{S}\colon g\mapsto\mathcal{S}(g) &{}=\int_M\tau_g^2 dvol_g\,,\\ \mathcal{F}_t\colon g\mapsto\mathcal{F}_t(g) &{}=\int\{\|\rho_g\|^2+t\tau_g^2\}dvol_g, \end{aligned} \] where $\rho$ and $\tau$ denote the Ricci tensor and the scalar curvature of the metric $g$. These quadratic curvature functionals have been extensively studied in mathematics and physics.

The aim of this talk is to show some results on the classification of Lorentzian critical metrics in the homogeneous setting. A specific feature of the Lorentzian situation is the existence of non-Einstein metrics which are critical for all quadratic curvature functionals.

Ricci solitons as submanifolds of complex hyperbolic spaces

Ángel Cidre Díaz (Universidade de Santiago de Compostela, Spain)

As a consequence of the recently solved Alekseevsky conjecture [1], and a result lying at the intersection of Ado's theorem for Lie algebras and the Nash embedding theorem [2], any expanding homogeneous Ricci soliton can be found, up to isometry, as a Lie subgroup of the solvable Iwasawa group associated with a symmetric space of non-compact type, considered with the induced metric. Motivated by this fact, we have addressed the classification of homogeneous Ricci solitons arising as Lie subgroups of the solvable Iwasawa groups of complex hyperbolic spaces. We also analyse the minimality of the examples obtained.

Bibliography:

1. C. Böhm, R. Lafuente. Non-compact Einstein manifolds with symmetry. J. Amer. Math. Soc. 36 (3), 591-651.
2. M. Jablonski. Einstein solvmanifolds as submanifolds of symmetric spaces. arXiv:1810.11077.

Trajectories of relativistic particles with curvature and torsion in generalized Robertson-Walker spacetimes

Martín de la Rosa Díaz (Universidad de Córdoba, Spain)

The dynamics of certain geometric functionals describing classical relativistic particles are studied in three-dimensional generalized Robertson-Walker (GRW) spacetimes. Such functionals depend on either the first or the second Frenet curvatures of the particle trajectory, in the vein of the so-called Plyushchay model [1]. In contrast with previous studies in GRW spacetimes [2], [3], in this work we consider both spacelike and timelike solutions (the latter being more realistic from a relativistic point of view). We establish interesting relationships between the curvature and torsion of critical curves and the curvature of the spatial fiber of the cosmological model, as well as conserved quantities whose physical significance is discussed.

Bibliography:

1. M. S. Plyushchay. Massive relativistic point particle with rigidity. International Journal of Modern Physics A 4, 15 (1989), 3851-3865.
2. Herrera, J., de la Rosa, M., and Rubio, R. M. Relativistic particles with torsion in three-dimensional non-vacuum spacetimes. Journal of Mathematical Physics 62 (2021), 062502.
3. Herrera, J., de la Rosa, M., and Rubio, R. M. On the dynamics of relativistic particles with torsion in warped product spacetimes. Journal of Physics A: Mathematical and Theoretical 55, 24 (2022), 245201.

Advances in the Classification of Conformally Einstein Lorentzian Lie Groups in Dimension 4

Ixchel Gutiérrez-Rodríguez (Universidade de Vigo, Spain)

Since Einstein's presentation of general relativity in 1915, the theory has experienced rapid success and development. This led to the emergence of various relativistic theories, including conformal gravity. In 1918, Hermann Weyl and Richard Bach developed a gravitational theory that incorporated the field equations of general relativity, allowing for the variation of a scale factor throughout spacetime. Additionally, they introduced the Bach tensor as a tool to characterize geometric properties of spacetime, which has found significant applications in theoretical physics and cosmology, \[ \mathfrak{B}=\operatorname{div}_2\operatorname{div}_4W+\tfrac{1}{2}W[\rho]. \]

In general relativity, a Lorentzian metric is seen as a gravitational potential that is related to the equation $\rho -\frac{1}{2}\tau g =T$. As a consequence, the interest in studying the Einstein manifolds arises, as the condition to describe a gravitational field in a vacuum $\rho = 0$. Conformal gravity refers to theories of gravity that exhibit invariance under conformal transformations, providing a framework to describe the structure of spacetime in terms of equivalence classes of metrics. As a result, the classification of Conformally Einstein metrics becomes an important problem in this field.

In this talk, we will delve into the classification of Conformal Einstein and Bach-flat Lorentzian metrics on the semi-direct extensions of Heisenberg $H_3\rtimes\mathbb{R}$, Euclidean $\widetilde{E}(2)\rtimes\mathbb{R}$ or Poincaré $E(1,1)\rtimes\mathbb{R}$ Lie groups in dimension four. This approach will provide new examples within the framework of conformal gravity.

Extensions in Lorentzian metric spaces

Jónatan Herrera Fernández (U. of Córdoba, Spain)

For decades, there has been an interest in obtaining a low regularity approximation to General Relativity. Several problems, such as the intrinsic study of causality, quantum field theory, or numerical approaches to General Relativity, require the study of discrete models.

Authors such as Kronheimer and Penrose, Bombelli, Lee, Meyer, and Sorkin, or Harris have conducted studies in this direction in the past, with a focus on the causal structure of spacetime. However, it is only recently that the theory has seen significant development, driven on one hand by the detection of gravitational waves and the precise observation of black holes, and on the other hand by the seminal work of Kunzinger and Sämmann [1] on Lorentzian Length Spaces.

This low-regularity approach requires the development of several well-known constructions and results from the differentiable case, including a natural notion of extension. In fact, a well-defined notion of extension will subsequently allow for the precise definition of concepts such as a black hole.

The aim of this presentation is twofold: First, we will introduce the notion of Lorentzian Metric Space, an analogue to the notion of classical metric space where distance is given by the so-called temporal separation. Its definition is obtained by considering core elements from the notion of Lorentzian Length Spaces (among others) in order to properly define a notion of extension. Secondly, we will explore how it is possible to extend the notion of causal completion [2] in this low regularity context, endowing this completion with a Lorentzian metric space structure. Finally, we will briefly analyze the connection between the global causality of the completion and its original space.

The presentation is based on the paper [3].

Bibliography:

1. M. Kunzinger and C. Sämann, Lorentzian length spaces, Annals of Global Analysis and Geometry, Vol. 54, No. 3, P. 399-447, 2008.
2. J. L. Flores, J. Herrera and M. Sánchez, On the final definition of the causal boundary and its relation with the conformal boundary, Adv. in Theor. and Math. Phys., 15, 991-1057, (2011).
3. S. Burgos, J. L. FLores, J. Herrera, The c-completion of Lorentzian metric spaces. arXiv:2305.02004

Einstein field equations in smooth metric measure spaces: Variational approach and solutions

Diego Mojón Álvarez (Universidade de Santiago de Compostela, Spain)

A semi-Riemannian manifold $(M,g)$ can be generalized by introducing a density function $f$ (equivalently, a distinguished measure), giving rise to a smooth metric measure space $(M,g, e^{-f} dvol_g)$. This change in the volume element affects any geometric object defined through integration, thus influencing the geometry of the underlying manifold. Hence, the geometric features of smooth metric measure spaces may be studied through so-called weighted tensors, which generalize the usual curvature-related tensors of a semi-Riemannian manifold while including information on the density function.

In this talk, we consider a Lorentzian smooth metric measure space and its associated Bakry-Émery Ricci tensor. We present a suitable generalization of the Einstein tensor to smooth metric measure spaces (called the weighted Einstein tensor) and a variational approach in order to obtain its associated field equations. These equations were proposed in [1] from a non-variational point of view. We consider the solutions to the resulting vacuum equations and observe that, as for usual Einstein metrics, the scalar curvature is constant for Einstein manifolds in this weighted sense. Finally, we go over some particular vacuum solutions, both in the isotropic ($\nabla f$ lightlike) and non-isotropic ($\nabla f$ timelike or spacelike) cases. In the first case, all solutions are realized on Brinkmann waves and Kundt spacetimes. In the second one, we have a wider range of geometries, so we introduce some curvature conditions in order to manage the problem. In particular, we present some rigidity results for 4-dimensional solutions with $\nabla f$ timelike and harmonic curvature tensor.

Bibliography:

1. M. Brozos-Vázquez and D. Mojón-Álvarez; Vacuum Einstein field equations in smooth metric measure spaces: the isotropic case. Class. Quantum Grav. 39 (13) (2022) 135013, 20 pp.

Null hypersurfaces, null cones and Killing horizons

Benjamín Olea Andrades (Universidad de Málaga, Spain)

Null hypersurfaces are an important family of hypersurfaces in a Lorentzian manifold. For example, black hole horizons and null cones (at least locally) are null hypersurfaces. The study of null hypersurfaces present some difficulties, being the main of them the degeneracy of the induced metric. In this talk, we focus on two independent problems which we address using the same technique.

The future null cone with vertex at a point $p$ is formed by the set of future null geodesics that emane from $p$ or, in other words, the image by the exponential map of the future tangent null cone at $p$. Near the vertex, a null cone is a null embedded hypersurface, but far away maybe is not due to the presence of null conjugate points or self-intersections. A null cone in a generalized Robertson-Walker space is totally umbilical, so we give conditions on a totally umbilical null hypersurface to be contained in null cone in an arbitrary Lorentzian manifold, under the null convergence condition. After that, we analyze the regularity of a null cone. Although a null conjugate point is a singular point of the exponential map, maybe the image by the exponential map (i.e. the null cone) can be endowed with the structure of embedded hypersurface. Indeed, there are easy examples of a map with a singular point but such that its image is an embedded hypersurface of the codomain. We show that this is not the case for the exponential map, i.e., a null cone can not be given a null hypersurface structure around a null conjugate point. A related problem, but in a Riemannian setting, was studied ninety years ago by Morse and Littauer.

On the other hand, a Killing horizon is a totally geodesic null hypersurfaces such that there is a Killing vector field which is tangent and null over the null hypersurface. Applying the same technique as before, we give conditions for a null hypersurface to be a Killing horizon.

Non-trivial k-contact structures

Xavier Rivas (Universidad Internacional de La Rioja, Spain)

The concepts of contact distribution and contact form are closely related but not entirely equivalent. While contact distributions can be locally described as the kernel of contact forms, the existence of a global contact form is not always guaranteed. Global contact forms are not unique either, since multiplying one by a non-vanishing function produces a new global contact one-form with the same contact distribution. This has significant implications, such as the alteration of the Reeb vector field. To address these issues, one can consider contact geometry as a form of homogeneous symplectic geometry as proved by Grabowski and Grabowska. In this presentation, we will review and extend their approach to encompass k-contact structures, which were recently introduced to provide geometric descriptions of non-conservative field theories.

Spacelike hypersurfaces in twisted product spacetimes with complete fiber and Calabi-Bernstein-type problems

Alberto Soria Marina (Universidad Politécnica de Madrid, Spain)

Product manifolds are of particular interest in the area of geometry and physics. Among these are the so-called twisted product spacetimes, which extend the Generalized Robertson Walker (GRW) ones, first introduced in 1995 by L. Alías, A. Romero and M. Sánchez, and which are of remarkable importance for their numerous applications in cosmology, among others. In this talk I focus on twisted product spacetimes of the form $I\times_f F$, where the fiber $(F,g_F)$ is a complete Riemannian manifold of arbitrary dimension. Some issues regarding global hyperbolicity will be addressed, and conditions for a twisted product spacetime to be an actual warped product will be shown. In the particular case where $(F,g_F)$ is closed (compact and without boundary) and the ambient spacetime has a suitable expanding behaviour, non-existence results for constant mean curvature hypersurfaces will be presented. Remarkable families of hypersurfaces will be characterized in this background and also in twisted product spacetimes of the form $I\,{ }_{f}\!\!\times F$ with a one-dimensional Lorentzian fiber. To conclude, some Calabi-Bernstein-type results will be put forward.

Contact formalism on Lie algebroids

Silvia Souto Pérez (Universidade de Santiago de Compostela, Spain)

The extension of Lagrangian and Hamiltonian formalisms to other geometric contexts, such as Lie algebroids, is already known and very useful in the case of symplectic systems. In this talk, we introduce a geometric description of contact Lagrangian and Hamiltonian systems on Lie algebroids using the notion of prolongation of a Lie algebroid over a mapping, introduced by Higgins and Mackenzie (1990). This has allowed us to define contact Lie algebroids, using the differential naturally associated to it, and therefore also the concept of a Legendrian Lie subalgebroid. We finally discuss the Hamilton-Jacobi problem in this framework.

Main reference

• A. Anahory Simoes, L. Colombo, M. de León, M. Salgado, S. Souto. Contact formalism on Lie algebroids, (in progress).

Complementary references

• A. Bravetti, H. Cruz, and D. Tapias. Contact Hamiltonian Mechanics. Annals of Physics, vol. 376, pp. 17-39.
• M. de León, J.C. Marrero, E. Martínez. Lagrangian submanifolds and dynamics on Lie algebroids. J. Phys. A: Math. Gen. 38 (2005), R241-R308.
• E. Martínez. Lagrangian mechanics on Lie algebroids. Acta Appl. Math. 67 (2001), no. 3, 295-320.

Shifted Courant-Nijenhuis structures

Paulo Antunes (Universidade de Coimbra, Portugal)

Given a vector bundle $A\to M$, shifted Courant algebroids on $A\oplus \wedge^{k-1}A^*$ are characterized as $Q$-structures on the (shifted) graded manifold $T^*[k]A[1]$ (see [2, 3, 4]). In this work we study Nijenhuis structures on shifted Courant algebroids and relate them with multisymplectic and Nambu-Poisson structures.

Bibliography:

1. Y. Bi, Y. Sheng, On higher analogues of Courant algebroids, Sci. China Math. 54 (2011), no. 3, 437-447.
2. N. Ikeda, K. Uchino, QP-structures of degree 3 and 4D topological field theory, Comm. Math. Phys. 303 (2011), no. 2, 317-330.
3. M. Cueca, The geometry of graded cotangent bundles, J. Geom. Phys. 161 (2021), paper no. 104055, 20 pp.
4. M. Zambon, $L_\infty$-algebras and higher analogues of Dirac structures and Courant algebroids, J. Symplectic Geom. 10 (2012), no. 4, 563-599.

Generalizing Euclidean splines to SPD manifolds

Luís Machado (ISR & UTAD, Portugal)

Symmetric positive definite (SPD) matrices are widely used in data science applications. Developing interpolation schemes for SPD matrices is clearly very important but can be quite demanding, since the most natural geometric framework of the SPD space is given by a Riemannian metric. The generalization of Euclidean cubic splines to Riemannian manifolds was driven by the need to solve trajectory planning problems for rigid body motions [5, 3]. A higher-order interpolation method in Riemannian manifolds, inspired by the optimizing properties of Euclidean splines, was introduced in [2] and gave rise to the so-called geometric splines.

Our goal is to present geometric splines in SPD manifolds, based on our recent work [1]. These manifolds will be equipped with the Log-Cholesky metric and the Lie group structure introduced in [5]. We first show how to derive a necessary and sufficient condition for a curve in SPD to be a geometric spline. We also present a closed form expression for cubic polynomials satisfying boundary conditions on position and velocity and obtain easy-to-compute expressions for higher order interpolation curves.

Acknowledgments

The work of Margarida Camarinha was partially supported by the Centre for Mathematics of the University of Coimbra - UIDB/00324/2020, funded by the Portuguese Government through FCT/MCTES. The work of Luís Machado and Fátima Silva Leite has been supported by Fundação para a Ciência e Tecnologia (FCT) under the project UIDP/00048/2020.

Bibliography:

1. Camarinha, M., Machado, L., Silva Leite, F.: k-Splines on SPD manifolds. Pré-publicações do Departamento de Matemática da Universidade de Coimbra, 23-14, 2023.
2. Camarinha, M., Silva Leite, F., Crouch, P.: Splines of class $C^k$ on non-Euclidean spaces. IMA J. Math. Control Inf. 12, 399-410 (1995).
3. Crouch, P., Silva Leite, F.: The dynamic interpolation problem on Riemannian manifolds, Lie groups and symmetric spaces. J. Dyn. Control Syst. 1 (2), 177-202 (1995).
4. Lin, Z.: Riemannian geometry of symmetric positive definite matrices via Cholesky decomposition. SIAM J. Matrix Anal. Appl. 40 (4), 1353-1370 (2019).
5. Noakes, L., Heinzinger, G., Paden, B.: Cubic splines on curved spaces. IMA J. Math. Control Inform. 6, 465-473 (1989).

New applications of the Ahlfors Laplacian: Ricci almost solitons and general relativistic vacuum constraint equations

Josef Mikes (Palacky University, Czech Republic)

In our report, we consider a ${L}^{2}$-orthogonal decomposition of the traceless part of the Ricci tensor of a closed Riemannian manifold and study its application to the geometry of compact Ricci almost solitons. In addition, we consider a ${L}^{2}$-orthogonal expansion of the traceless part of the second fundamental form of a closed space-like hypersurface in a Lorentzian manifold and study its application to the problem of constructing solutions of general relativistic vacuum constraint equations. In these two cases, we use the well-known Ahlfors Laplacian.

Hamilton-Jacobi theory for contact Lagrangian systems

Miguel C. Muñoz Lecanda (Universitat Politècnica de Catalunya, Spain)

We present a new framework for the Hamilton-Jacobi theory for non-conservative Lagrangian mechanical systems, which is based on the contact formulation for these kinds of systems. Several types of dynamical equations can be considered for these systems and their corresponding Hamilton-Jacobi equations are studied. Some examples are also analyzed.

Multicontact formulation of field theories and applications to Physics

Narciso Román Roy (UPC, Spain)

A new geometric structure inspired by multisymplectic and contact geometries, called multicontact structure, has been developed recently to describe non-conservative and action-dependent classical field theories. We review the main features of this formulation and how it is applied to study some classical theories in theoretical physics which are modified in order to include action-dependence; in particular: action-dependent quadratic and affine Lagrangian theories in general, unidimensional wave equation with dissipation, modified Klein-Gordon equation, action-dependent bosonic string, and Burgers' equation from the heat equation, among others.