Riemannian Flow Research Articles

Covariant bit threads

We derive several new reformulations of the Hubeny-Rangamani-Takayanagi covariant holographic entanglement entropy formula. These include: (1) a minimax formula, which involves finding a maximal-area achronal surface on a timelike hypersurface homologous to D(A) (the boundary causal domain of the region A whose entropy we are calculating) and minimizing over the hypersurface; (2) a max V-flow formula, in which we maximize the flux through D(A) of a divergenceless bulk 1-form V subject to an upper bound on its norm that is non-local in time; and (3) a min U-flow formula, in which we minimize the flux over a bulk Cauchy slice of a divergenceless timelike 1-form U subject to a lower bound on its norm that is non-local in space. The two flow formulas define convex programs and are related to each other by Lagrange duality. For each program, the optimal configurations dynamically find the HRT surface and the entanglement wedges of A and its complement. The V-flow formula is the covariant version of the Freedman-Headrick bit thread reformulation of the Ryu-Takayanagi formula. We also introduce a measure-theoretic concept of a “thread distribution”, and explain how Riemannian flows, V-flows, and U-flows can be expressed in terms of thread distributions.

Sewing spacetime with Lorentzian threads: complexity and the emergence of time in quantum gravity

Holographic entanglement entropy was recently recast in terms of Riemannian flows or ‘bit threads’. We consider the Lorentzian analog to reformulate the ‘complexity=volume’ conjecture using Lorentzian flows — timelike vector fields whose minimum flux through a boundary subregion is equal to the volume of the homologous maximal bulk Cauchy slice. By the nesting of Lorentzian flows, holographic complexity is shown to obey a number of properties. Particularly, the rate of complexity is bounded below by conditional complexity, describing a multi-step optimization with intermediate and final target states. We provide multiple explicit geometric realizations of Lorentzian flows in AdS backgrounds, including their time-dependence and behavior near the singularity in a black hole interior. Conceptually, discretized flows are interpreted as Lorentzian threads or ‘gatelines’. Upon selecting a reference state, complexity thence counts the minimum number of gatelines needed to prepare a target state described by a tensor network discretizing the maximal volume slice, matching its quantum information theoretic definition. We point out that suboptimal tensor networks are important to fully characterize the state, leading us to propose a refined notion of complexity as an ensemble average. The bulk symplectic potential provides a specific ‘canonical’ thread configuration characterizing perturbations around arbitrary CFT states. Consistency of this solution requires the bulk satisfy the linearized Einstein’s equations, which are shown to be equivalent to the holographic first law of complexity, thereby advocating for a principle of ‘spacetime complexity’. Lastly, we argue Lorentzian threads provide a notion of emergent time. This article is an expanded and detailed version of [1], including several new results.

The Bochner Formula for Riemannian Flows

On a Riemannian manifold (M, g) endowed with a Riemannian flow, we study in this paper the curvature term in the Bochner–Weitzenböck formula of the basic Laplacian. We prove that this term splits into two parts; a first part that depends on the curvature operator of the manifold M and a second part that can be expressed in terms of the O’Neill tensor of the flow. After getting a lower bound for this curvature term depending on a bound of each of these two parts, we establish an eigenvalue estimate for the basic Laplacian. We then discuss the limiting case of this latter estimate and prove that, when equality occurs, the manifold M is isometric to a local product.

The Hessian Riemannian flow and Newton’s method for effective Hamiltonians and Mather measures

Effective Hamiltonians arise in several problems, including homogenization of Hamilton–Jacobi equations, nonlinear control systems, Hamiltonian dynamics, and Aubry–Mather theory. In Aubry–Mather theory, related objects, Mather measures, are also of great importance. Here, we combine ideas from mean-field games with the Hessian Riemannian flow to compute effective Hamiltonians and Mather measures simultaneously. We prove the convergence of the Hessian Riemannian flow in the continuous setting. For the discrete case, we give both the existence and the convergence of the Hessian Riemannian flow. In addition, we explore a variant of Newton’s method that greatly improves the performance of the Hessian Riemannian flow. In our numerical experiments, we see that our algorithms preserve the non-negativity of Mather measures and are more stable than related methods in problems that are close to singular. Furthermore, our method also provides a way to approximate stationary MFGs.

Spatially Regularized Geometric Assignment for Unsupervised Label Learning on Manifolds

AbstractWe introduce a smooth coupled system of Riemannian flows for simultaneously learning a dictionary of manifold‐valued prototypes and assigning these prototypes to the image data in a spatially coherent way. In particular, we detail a numerical update scheme for the evolution of prototypes on the Riemannian manifold of symmetric positive definite matrices. The scheme is applied to covariance descriptors are extracted from image data in order to demonstrate our approach.

Ricci–Yamabe maps for Riemannian flows and their volume variation and volume entropy

The aim of this short note is to produce new examples of geometrical flows associated to a given Riemannian flow $g(t)$. The considered flow in covariant symmetric $2$-tensor fields will be called Ricci-Yamabe map since it involves a scalar combination of Ricci tensor and scalar curvature of $g(t)$. Due to the signs of considered scalars the Ricci-Yamabe flow can be also a Riemannian or semi-Riemannian or singular Riemannian flow. We study the associated function of volume variation as well as the volume entropy. Finally, since the two-dimensional case was the most commonly addressed situation we express the Ricci flow equation in all four orthogonal separable coordinate systems of the plane.

Uniqueness of Viscosity Mean Curvature Flow Solution in Two Sub-Riemannian Structures

We provide a uniqueness result for a class of viscosity solutions to sub-Riemannian mean curvature flows. In a sub-Riemannian setting, uniqueness cannot be deduced by the comparison principle, which is known only for graphs and for radially symmetry surfaces. Here we use a definition of continuous viscosity solutions of sub-Riemannian mean curvature flows motivated from a regularized Riemannian approximation of the flow. With this definition, we prove that any continuous viscosity solution of the equation is a limit of a sequence of solutions of Riemannian flow and obtain as a consequence uniqueness and the comparison principle. The results are provided in the settings of both 3-dimensional rototranslation group ${SE}(2)$ and Carnot groups of step 2, which are particularly important due to their relation to the surface completion problem of a model of the visual cortex.

On the complete integrability of the geodesic flow of pseudo-H-type Lie groups

Pseudo-H-type groups $$G_{r,s}$$ form a class of step-two nilpotent Lie groups with a natural pseudo-Riemannian metric. In this paper the question of complete integrability in the sense of Liouville is studied for the corresponding (pseudo-)Riemannian geodesic flow. Via the isometry group of $$G_{r,s}$$ families of first integrals are constructed. A modification of these functions gives a set of $$\dim G_{r,s}$$ functionally independent smooth first integrals in involution. The existence of a lattice L in $$G_{r,s}$$ is guaranteed by recent work of K. Furutani and I. Markina. The complete integrability of the pseudo-Riemannian geodesic flow of the compact nilmanifold $$L \backslash G_{r,s}$$ is proved under additional assumptions on the group $$G_{r,s}$$ .

Singular Riemannian flows and characteristic numbers

Let $M$ be an even-dimensional, oriented closed manifold. We show that the restriction of a singular Riemannian flow on $M$ to a small tubular neighborhood of each connected component of its singular stratum is foliated-diffeomorphic to an isometric flow on the same neighborhood. We then prove a formula that computes characteristic numbers of $M$ as the sum of residues associated to the infinitesimal foliation at the components of the singular stratum of the flow.

Eigenvalue Estimate for the Basic Laplacian on Manifolds with Foliated Boundary, Part II

On a compact Riemannian manifold N whose boundary is endowed with a Riemannian flow, we gave in El Chami et al. (Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary, 2015) a sharp lower bound for the first non-zero eigenvalue of the basic Laplacian acting on basic 1-forms. In this paper, we extend this result to the set of basic p-forms when $$p>1$$ . We then characterize the limiting case by showing that the manifold N is isometric to for some group $$\Gamma $$ where $$B'$$ denotes the unit closed ball. As a consequence, we describe the Riemannian product $${\mathbb {S}}^1\times {\mathbb {S}}^n$$ as the boundary of a manifold.

Riemannian flows and adiabatic limits

We show the convergence properties of the eigenvalues of the Dirac operator on a spin manifold with a Riemannian flow when the metric is collapsed along the flow.

Variations of the total mixed scalar curvature of a distribution

We examine the total mixed scalar curvature of a smooth manifold endowed with a distribution as a functional of a pseudo-Riemannian metric. We develop variational formulas for quantities of extrinsic geometry of the distribution and use this key and technical result to find the critical points of this action. Together with the arbitrary variations of the metric, we consider also variations that preserve the volume of the manifold or partially preserve the metric (e.g., on the distribution). For each of those cases, we obtain the Euler–Lagrange equation and its several solutions. Examples of critical metrics that we find are related to various fields of geometry such as contact and 3-Sasakian manifolds, geodesic Riemannian flows, codimension-one foliations, and distributions of interesting geometric properties (e.g., totally umbilical and minimal).

Eigenvalue estimate for the basic Laplacian on manifolds with foliated boundary

On a compact Riemannian manifold whose boundary is endowed with a Riemannian flow, we give a sharp lower bound for the first eigenvalue of the basic Laplacian acting on basic 1-forms. The equality case gives rise to a particular geometry of the flow and of the boundary. Namely, we prove that the flow is a local product and the boundary is $$\eta $$ -umbilical. This allows to characterize the quotient of $${\mathbb {R}}\times B'$$ by some group $$\Gamma $$ as the limiting manifold. Here $$B'$$ denotes the unit closed ball. Finally, we deduce several rigidity results describing the product $${\mathbb {S}}^1\times {\mathbb {S}}^n$$ as the boundary of a manifold.

Rigidity Results for Riemannian Spin $$^c$$ c Manifolds with Foliated Boundary

Given a Riemannian spin $$^c$$ manifold whose boundary is endowed with a Riemannian flow, we show that any solution of the basic Dirac equation satisfies an integral inequality depending on geometric quantities, such as the mean curvature and the O’Neill tensor. We then characterize the equality case of the inequality when the ambient manifold is a domain of a Kahler–Einstein manifold or a Riemannian product of a Kahler–Einstein manifold with $$\mathbb R$$ (or with the circle $$\mathbb S^1$$ ).

A gradient-type deformation of conics and a class of Finslerian flows

Abstract The aim of this paper is to produce new examples of Riemannian and Finsler structures having as model a scalar deformation of conics inspired by the scaling transformation. It continues [4] from the point of view of relationship between quadratic polynomials (which provide equations of conics in dimension 2) and Finsler geometries. A type of Finslerian ow is introduced, based on the previous deformation and we completely solve the corresponding particular case of Riemannian flow.

Tzitzeica equations and Tzitzeica surfaces in separable coordinate systems and the Ricci flow tensor field

The Tzitzeica equation and two well-known Tzitzeica surfaces are studied in the separable coordinate systems on the plane and space respectively. We study also Tzitzeica graphs with a parameter and interpret the induced class of first fundamental forms as a Riemannian flow. Consequently, we introduce a tensor field which measures how far is a given Riemannian flow to be a Ricci one. This tensor field is explicitly computed for the case of a initial isothermic metric and a flow of convex type.

Riemannian and Sub-Riemannian Geodesic Flows

We show that the geodesic flows of a sub-Riemannian metric and that of a Riemannian extension commute if and only if the extended metric is parallel with respect to a certain connection. This result allows us to describe the sub-Riemannian geodesic flow on totally geodesic Riemannian foliations in terms of the Riemannian geodesic flow. Also, given a submersion $$\pi :M \rightarrow B$$ , we describe when the projections of a Riemannian and a sub-Riemannian geodesic flow in M coincide.

Rigidity results for spin manifolds with foliated boundary

In this paper, we consider a compact Riemannian manifold whose boundary is endowed with a Riemannian flow. Under a suitable curvature assumption depending on the O’Neill tensor of the flow, we prove that any solution of the basic Dirac equation is the restriction of a parallel spinor field defined on the whole manifold. As a consequence, we show that the flow is a local product. In particular, in the case where solutions of the basic Dirac equation are given by basic Killing spinors, we characterize the geometry of the manifold and the flow.

Tenseness of Riemannian flows

We show that any transversally complete Riemannian foliation eF&/em& of dimension one on any possibly non-compact manifold M is tense; namely, (M,eF&/em&) admits a Riemannian metric such that the mean curvature form of eF&/em& is basic. This is a partial generalization of a result of Dominguez, which says that any Riemannian foliation on any compact manifold is tense. Our proof is based on some results of Molino and Sergiescu, and it is simpler than the original proof by Dominguez. As an application, we generalize some well known results including Masa's characterization of tautness.

Deformation of Sasakian metrics

Deformations of the Reeb flow of a Sasakian manifold as transversely K\"ahler flows may not admit compatible Sasakian metrics anymore. We show that the triviality of the (0,2)-component of the basic Euler class characterizes the existence of compatible Sasakian metrics for given small deformations of the Reeb flow as transversely holomorphic Riemannian flows. We also prove a Kodaira-Akizuki-Nakano type vanishing theorem for basic Dolbeault cohomology of homologically orientable transversely K\"ahler foliations. As a consequence of these results, we show that any small deformations of the Reeb flow of a positive Sasakian manifold admit compatible Sasakian metrics.