Sectional curvature
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Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries Open
Let $\M$ be a smooth connected manifold endowed with a smooth measure $\mu$ and a smooth locally subelliptic diffusion operator $L$ satisfying $L1=0$, and which is symmetric with respect to $\mu$. Associated with $L$ one has \textitle carr…
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Ricci–Yamabe maps for Riemannian flows and their volume variation and volume entropy Open
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 scala…
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An extension of a theorem of Wu–Yau Open
We show that a compact Kähler manifold with nonpositive holomorphic sectional curvature has nef canonical bundle. If the holomorphic sectional curvature is negative then it follows that the canonical bundle is ample, confirming a conjectur…
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Principal component analysis for functional data on Riemannian manifolds and spheres Open
Functional data analysis on nonlinear manifolds has drawn recent interest. Sphere-valued functional data, which are encountered, for example, as movement trajectories on the surface of the earth are an important special case. We consider a…
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Curvature: A Variational Approach Open
The curvature discussed in this paper is a far reaching generalisation of the Riemannian sectional curvature. We give a unified definition of curvature which applies to a wide class of geometric structures whose geodesics arise from optima…
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Conformal $ \eta $-Ricci solitons within the framework of indefinite Kenmotsu manifolds Open
The present paper is to deliberate the class of $ \epsilon $-Kenmotsu manifolds which admits conformal $ \eta $-Ricci soliton. Here, we study some special types of Ricci tensor in connection with the conformal $ \eta $-Ricci soliton of $ \…
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On real bisectional curvature for Hermitian manifolds Open
Motivated by the recent work of Wu and Yau on the ampleness of a canonical line bundle for projective manifolds with negative holomorphic sectional curvature, we introduce a new curvature notion called real bisectional curvature for Hermit…
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Effective versions of the positive mass theorem Open
The study of stable minimal surfaces in Riemannian $3$-manifolds $(M, g)$ with non-negative scalar curvature has a rich history. In this paper, we prove rigidity of such surfaces when $(M, g)$ is asymptotically flat and has horizon boundar…
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On Jacobi fields and a canonical connection in sub-Riemannian geometry Open
In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first in…
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Yamabe solitons on three-dimensional Kenmotsu manifolds Open
Let the Riemannian metric of a three-dimensional Kenmotsu manifold be a Yamabe soliton. In this paper, we prove that the Kenmotsu manifold is of constant sectional curvature $-1$ and the Yamabe soliton is expanding with the soliton constan…
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Introducing quantum Ricci curvature Open
Motivated by the search for geometric observables in nonperturbative quantum\ngravity, we define a notion of coarse-grained Ricci curvature. It is based on a\nparticular way of extracting the local Ricci curvature of a smooth Riemannian\nm…
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Kähler manifolds of semi-negative holomorphic sectional curvature Open
In an earlier work, we investigated some consequences of the existence of a Kähler metric of negative holomorphic sectional curvature on a projective manifold. In the present work, we extend our results to the case of semi-negative (i.e., …
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The porous medium equation with large initial data on negatively curved\n Riemannian manifolds Open
We show existence and uniqueness of very weak solutions of the Cauchy problem\nfor the porous medium equation on Cartan-Hadamard manifolds satisfying suitable\nlower bounds on Ricci curvature, with initial data that can grow at infinity at…
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Control of eigenfunctions on surfaces of variable curvature Open
We prove a microlocal lower bound on the mass of high energy eigenfunctions of the Laplacian on compact surfaces of negative curvature, and more generally on surfaces with Anosov geodesic flows. This implies controllability for the Schrödi…
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Gap phenomena and curvature estimates for conformally compact Einstein manifolds Open
In this paper we obtain first a gap theorem for a class of conformally compact Einstein manifolds with a renormalized volume that is close to its maximum value. We also use a blow-up method to derive curvature estimates for conformally com…
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On the Heat Diffusion for Generic Riemannian and Sub-Riemannian Structures Open
In this paper we provide the small-time heat kernel asymptotics at the cut\nlocus in three relevant cases: generic low-dimensional Riemannian manifolds,\ngeneric 3D contact sub-Riemannian manifolds (close to the starting point) and\ngeneri…
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Sub-Riemannian Mean Curvature Flow for Image Processing Open
In this paper we reconsider the sub-Riemannian cortical model of image completion introduced in [G. Citti and A. Sarti, J. Math. Imaging Vision, 24 (2006), pp. 307–326]. This model combines two mechanisms, the sub-Riemannian diffusion and …
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Remarks on critical metrics of the scalar curvature and volume functionals on compact manifolds with boundary Open
We provide a general B\\"ochner type formula which enables us to prove some\nrigidity results for $V$-static spaces. In particular, we show that an\n$n$-dimensional positive static triple with connected boundary and positive\nscalar curvat…
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Curvature Invariants for Statistical Submanifolds of Hessian Manifolds of Constant Hessian Curvature Open
We consider statistical submanifolds of Hessian manifolds of constant Hessian curvature. For such submanifolds we establish a Euler inequality and a Chen-Ricci inequality with respect to a sectional curvature of the ambient Hessian manifol…
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Geometry, Analysis and Dynamics on sub-Riemannian Manifolds Open
peer reviewed
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Transverse Weitzenböck formulas and curvature dimension inequalities on Riemannian foliations with totally geodesic leaves Open
We prove a family of Weitzenböck formulas on a Riemannian foliation with totally geodesic leaves.These Weitzenböck formulas are naturally parametrized by the canonical variation of the metric.As a consequence, under natural geometric condi…
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Volume of small balls and sub-Riemannian curvature in 3D contact manifolds Open
We compute the asymptotic expansion of the volume of small subẊRiemannian balls in a contact 3-dimensional manifold, and we express the first meaningful geometric coefficients in terms of geometric invariants of the sub-Riemannian structur…
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Towards Riemannian Accelerated Gradient Methods Open
We propose a Riemannian version of Nesterov's Accelerated Gradient algorithm (RAGD), and show that for geodesically smooth and strongly convex problems, within a neighborhood of the minimizer whose radius depends on the condition number as…
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Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries Open
Let \mathbb M be a smooth connected manifold endowed with a smooth measure \mu and a smooth locally subelliptic diffusion operator L satisfying L1=0 , and which is symmetric with respect to \mu . Associated with L one has the carré du cham…
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Ollivier-Ricci curvature convergence in random geometric graphs Open
Connections between continuous and discrete worlds tend to be elusive. One example is curvature. Even though there exist numerous nonequivalent definitions of graph curvature, none is known to converge in any limit to any traditional defin…
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On warped product gradient η-Ricci solitons Open
If the potential vector field of an ?-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. In a particular case of irrotational potential vector field we prove that the…
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Sharp inequalities involving the Ricci curvature for Riemannian submersions Open
In this paper, we obtain sharp inequalities on Riemannian manifolds admitting a Riemannian submersion and give some characterizations using these inequalities. We improve Chen-Ricci inequality for Riemannian submersion and present some exa…
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Separability in Riemannian Manifolds Open
An outline of the basic Riemannian structures underlying the separation of variables in the Hamilton-Jacobi equation of natural Hamiltonian systems.
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Secondary large-scale index theory and positive scalar curvature Open
We develop a theory of secondary invariants associated to complete Riemannian metrics of uniformly positive scalar curvature outside a prescribed subset on a spin manifold. \nWe work in the context of large-scale (or "coarse") index theory…
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Nonsmooth differential geometry– An approach tailored for spaces with Ricci curvature bounded from below Open
We discuss in which sense general metric measure spaces possess a first order differential structure. Building on this, we then see that on spaces with Ricci curvature bounded from below a second order calculus can be developed, permitting…