arXiv subjects

André Magalhães de Sá Gomes

Publications and source records attributed to André Magalhães de Sá Gomes.

6 recordsLinked to original sources

arXiv2026

The Riemannian geometry of the probability space of the unit circle

This paper explores the Riemannian geometry of the Wasserstein space of the circle, namely $P(S^{1})$, the set of probability measures on the unit circle endowed with the 2-Wasserstein metric. Building on the foundational work of Otto, Lott, and Villani, the authors developed in another work an intrinsic framework for studying the differential geometry of Wasserstein spaces of compact Lie groups, making use of the Peter-Weyl Theorem. This formalism allowed them to explicit an example in this paper. Key contributions include explicit computations of the Riemannian metric matrix coefficients, Lie brackets, and the Levi-Civita connection, along with its associated Christoffel symbols. The geodesic equations and curves with constant velocity fields are analysed, expliciting their PDEs. Notably, the paper demonstrates that $P(S^{1})$ is flat, with vanishing curvature. These results provide a comprehensive geometric understanding of $P(S^{1})$, connecting optimal transport theory and differential geometry, with potential applications in dynamical systems.

math.DG↗

arXiv2026

On Differential and Riemannian Calculus on Wasserstein Spaces

In this paper we develop an intrinsic formalism to study the topology, smooth structure, and Riemannian geometry of the Wasserstein space of a closed Riemannian manifold. Our formalism allows for a new characterisation of the Weak topology via convergent sequences of the subjacent space. Applying it we also provide a new proof that Wasserstein spaces of closed manifolds are geodesically convex. Our framework is particularly handy to address the Wasserstein spaces of compact Lie groups, where we refine our formalism and present an explicit example.

math.DG↗

arXiv2026

The Wasserstein space of the circle is not flat

We show that the Wasserstein space of the circle is not flat, in contrast with the flatness of the Wasserstein spaces of the real line and of an interval. Its sectional curvature is governed by a symplectic form on the Fourier coefficients, so that a plane is flat exactly when it is isotropic for that form, and on the plane spanned by the $n$-th Fourier modes the curvature is bounded above by $3n^2$, with equality exactly at the uniform measure. It is therefore unbounded. We compute it inside the formal Riemannian structure of Otto and confirm it directly from the definition of $W_2$, which also shows that no neighbourhood of the uniform measure in $P_2(S^1)$ is CAT(0). The result corrects the widely stated expectation that a Wasserstein space is flat whenever its base manifold is one-dimensional. The computation is carried out in the global frame of the tangent bundle of $P^\infty(M)$ induced by the eigenfunctions of the Laplacian of $M$, whose existence was observed by Lott. We prove that it is a Schauder basis of every tangent space and write Otto's metric and its Levi-Civita connection in it. On the circle this gives the metric and connection coefficients of $P^\infty(S^1)$ in closed form, as finite combinations of the Fourier coefficients of the density. A final section puts the frame to work on a computation from dynamics, reading the derivative of the push-forward action of an expanding circle map at its invariant measure as a weighted shift on the Fourier frequencies.

math.DG↗

arXiv2026

Pushforward dynamics on Wasserstein spaces and measure rigidity

For an endomorphism $ϕ$ of a closed Riemannian manifold $M$, we study the pushforward action $ϕ_\ast$ on the Wasserstein space $\mathcal{P}(M)$ at a measure $μ_0$ preserved by $ϕ$. We show that if $ϕ$ is a $C^2$ covering map and $μ_0$ has positive $C^1$ density, then $ϕ_\ast$ is Gâteaux differentiable at $μ_0$ along tangent directions, with derivative given by the transfer operator of $ϕ$ acting on vector fields, followed by orthogonal projection onto the tangent space. The derivative is the adjoint of the Koopman operator restricted to the tangent space, and its fixed space consists of the directions in which $μ_0$ can be deformed while preserving invariance to first order. For appropriate pairs of endomorphisms of $\mathbb{T}^d$, we compute the intersection of their fixed spaces, show that it contains an infinite family of linearly independent continuous vector fields, and construct, for every $n$, an embedded $n$-dimensional family of measures that are nearly invariant under both endomorphisms. First-order rigidity therefore fails in every dimension. In particular, rigidity phenomena such as higher-dimensional analogues of the Furstenberg conjecture, if true, are genuinely nonlinear.

math.DS↗

arXiv2026

Displacement convexity of Invariant Measures and curvature bounds of isometric actions

Let $G\curvearrowright M$ be a proper isometric action of a compact Lie group on a complete, connected and orientable Riemannian manifold of dimension $N$. We characterize the local $K$-displacement convexity of the internal-energy functional $H$ on the space $\mathcal P^{ac}_G(M)$ of absolutely continuous $G$-invariant probability measures. Via disintegration along the principal orbits, $H$ reduces to the internal energy of a transversal density against the orbit-volume--weighted measure $\mathfrak m = V\operatorname{vol}_{M/G}$, and the convexity of $H$ is equivalent to the $N$-Bakry--Émery condition $\operatorname{Ric}^Ψ_N \ge K$ on the weighted quotient $(M/G,\mathfrak m)$. Written on $M$, this bound reads $\operatorname{Ric}^{\mathcal H}_M(v) + 3\|A_v\|^2 - \operatorname{Hess}(\log V)(v,v) - \langle\vec H,v\rangle^2/m \ge K|v|^2$ at every principal point and horizontal direction $v$, where $A$ is the O'Neill integrability tensor of $π\colon M\to M/G$, $\vec H$ the mean-curvature vector of the orbits, and $m=\dim(G\cdot x)$ the orbit dimension. The terms on the left encode, in this order, the horizontal curvature of $M$, the non-integrability of the horizontal distribution, and --- in the last two --- the variation of the orbit volume. When the orbits are points one recovers the theorem of von Renesse--Sturm.

math.DG↗

arXiv2021

Representations of low copolarity

We classify irreducible representations of compact connected Lie groups whose orbit space is isometric to the orbit space of a representation of a compact Lie group of dimension~$7$, $8$ or $9$. They turn out to be closely related to symmetric spaces, with one exception only.

math.RT↗