Book•
Nonsmooth Differential Geometry: An Approach Tailored for Spaces With Ricci Curvature Bounded from Below
31 Dec 2017-
TL;DR: In this article, it was shown that Hessian, covariant/exterior derivatives and Ricci curvature bounded from below a second-order calculus can be defined for general metric measure spaces.
Abstract: 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 to define Hessian, covariant/exterior derivatives and Ricci curvature.
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01 Jan 2018
TL;DR: In this paper, a non-collapsed space with Ricci curvature bounded from below is defined, and the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate are proved.
Abstract: We propose a definition of non-collapsed space with Ricci curvature bounded from below and we prove the versions of Colding's volume convergence theorem and of Cheeger-Colding dimension gap estimate for ${\sf RCD}$ spaces.
In particular this establishes the stability of non-collapsed spaces under non-collapsed Gromov-Hausdorff convergence.
111 citations
TL;DR: In this paper, an analogue of DiPerna-Lions theory on well-posedness of flows of ODEs associated to Sobolev vector fields is established.
Abstract: We establish, in a rather general setting, an analogue of DiPerna–Lions theory on well-posedness of flows of ODEs associated to Sobolev vector fields. Key results are a well-posedness result for the continuity equation associated to suitably defined Sobolev vector fields, via a commutator estimate, and an abstract superposition principle in (possibly extended) metric measure spaces, via an embedding into ℝ∞. When specialized to the setting of Euclidean or infinite-dimensional (e.g., Gaussian) spaces, large parts of previously known results are recovered at once. Moreover, the class of RCD(K,∞) metric measure spaces, introduced by Ambrosio, Gigli and Savare [Duke Math. J. 163:7 (2014) 1405–1490] and the object of extensive recent research, fits into our framework. Therefore we provide, for the first time, well-posedness results for ODEs under low regularity assumptions on the velocity and in a nonsmooth context.
93 citations
Posted Content•
TL;DR: In this article, it was shown that the push forward of the reference measure under the charts built by them is continuous with respect to the Lebesgue measure, which has relevant implications on the structure of tangent spaces to RCD$ spaces.
Abstract: Mondino and Naber recently proved that finite dimensional $\sf RCD$ spaces are rectifiable. Here we show that the push-forward of the reference measure under the charts built by them is absolutely continuous with respect to the Lebesgue measure. This result, read in conjunction with another recent work of us, has relevant implications on the structure of tangent spaces to $\sf RCD$ spaces. A key tool that we use is a recent paper by De Philippis-Rindler about the structure of measures on the Euclidean space.
69 citations
Cites background from "Nonsmooth Differential Geometry: An..."
...All these definitions and properties can be found in [9]....
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...On the other hand, for general ‘irregular’ spaces the approach in [9] has little to do with tangent spaces arising as pmGH-limits....
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...This identity and the locality of the differential (see [9]) imply that χEdf = χEd(g ◦ φ) so that taking into account the first in (2....
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...In [9], following some ideas of Weaver [18], it has been proposed an abstract definition of tangent ‘bundle’ to a metric measure space based on the properties of Sobolev functions....
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...This is relevant because it opens up the possibility of studying the ‘concrete and geometric’ notion of tangent space as pmGH-limit via the ‘abstract and analytic’ one proposed in [9]....
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TL;DR: In this paper, it was shown that volume cone implies metric cone in Ricci-limit spaces, thus generalising to this class of spaces a well known result of Cheeger-Colding.
Abstract: We prove that ‘volume cone implies metric cone’ in the setting of $${\mathsf{RCD}}$$
spaces, thus generalising to this class of spaces a well known result of Cheeger–Colding valid in Ricci-limit spaces.
56 citations
Posted Content•
TL;DR: In this article, a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces was proved for a newly defined quasi-metric built from the Green function of the Laplacian.
Abstract: We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from the Green function of the Laplacian. Its main application is that RCD(K,N) spaces have constant dimension. In this way we generalize to such abstract framework a result proved by Colding-Naber for Ricci limit spaces, introducing ingredients that are new even in the smooth setting.
38 citations
References
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Book•
01 Jan 1971
TL;DR: In this article, the authors present a table of abstractions for categories, including Axioms for Categories, Functors, Natural Transformations, and Adjoints for Preorders.
Abstract: I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large Categories.- 8. Hom-sets.- II. Constructions on Categories.- 1. Duality.- 2. Contravariance and Opposites.- 3. Products of Categories.- 4. Functor Categories.- 5. The Category of All Categories.- 6. Comma Categories.- 7. Graphs and Free Categories.- 8. Quotient Categories.- III. Universals and Limits.- 1. Universal Arrows.- 2. The Yoneda Lemma.- 3. Coproducts and Colimits.- 4. Products and Limits.- 5. Categories with Finite Products.- 6. Groups in Categories.- IV. Adjoints.- 1. Adjunctions.- 2. Examples of Adjoints.- 3. Reflective Subcategories.- 4. Equivalence of Categories.- 5. Adjoints for Preorders.- 6. Cartesian Closed Categories.- 7. Transformations of Adjoints.- 8. Composition of Adjoints.- V. Limits.- 1. Creation of Limits.- 2. Limits by Products and Equalizers.- 3. Limits with Parameters.- 4. Preservation of Limits.- 5. Adjoints on Limits.- 6. Freyd's Adjoint Functor Theorem.- 7. Subobjects and Generators.- 8. The Special Adjoint Functor Theorem.- 9. Adjoints in Topology.- VI. Monads and Algebras.- 1. Monads in a Category.- 2. Algebras for a Monad.- 3. The Comparison with Algebras.- 4. Words and Free Semigroups.- 5. Free Algebras for a Monad.- 6. Split Coequalizers.- 7. Beck's Theorem.- 8. Algebras are T-algebras.- 9. Compact Hausdorff Spaces.- VII. Monoids.- 1. Monoidal Categories.- 2. Coherence.- 3. Monoids.- 4. Actions.- 5. The Simplicial Category.- 6. Monads and Homology.- 7. Closed Categories.- 8. Compactly Generated Spaces.- 9. Loops and Suspensions.- VIII. Abelian Categories.- 1. Kernels and Cokernels.- 2. Additive Categories.- 3. Abelian Categories.- 4. Diagram Lemmas.- IX. Special Limits.- 1. Filtered Limits.- 2. Interchange of Limits.- 3. Final Functors.- 4. Diagonal Naturality.- 5. Ends.- 6. Coends.- 7. Ends with Parameters.- 8. Iterated Ends and Limits.- X. Kan Extensions.- 1. Adjoints and Limits.- 2. Weak Universality.- 3. The Kan Extension.- 4. Kan Extensions as Coends.- 5. Pointwise Kan Extensions.- 6. Density.- 7. All Concepts are Kan Extensions.- Table of Terminology.
9,254 citations
Book•
01 Jan 1979
3,776 citations
"Nonsmooth Differential Geometry: An..." refers background in this paper
...11 is that it shows that a posteriori the theory of separable Hilbert modules and that of direct integral of Hilbert spaces (a concept generalizing that of direct sum to a ‘continuous family of indexes’ - we refer to [55] for an overview of this topic and detailed bibliography) are tightly linked....
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Book•
01 Jan 2005
TL;DR: In this article, Gradient flows and curves of Maximal slopes of the Wasserstein distance along geodesics are used to measure the optimal transportation problem in the space of probability measures.
Abstract: Notation.- Notation.- Gradient Flow in Metric Spaces.- Curves and Gradients in Metric Spaces.- Existence of Curves of Maximal Slope and their Variational Approximation.- Proofs of the Convergence Theorems.- Uniqueness, Generation of Contraction Semigroups, Error Estimates.- Gradient Flow in the Space of Probability Measures.- Preliminary Results on Measure Theory.- The Optimal Transportation Problem.- The Wasserstein Distance and its Behaviour along Geodesics.- Absolutely Continuous Curves in p(X) and the Continuity Equation.- Convex Functionals in p(X).- Metric Slope and Subdifferential Calculus in (X).- Gradient Flows and Curves of Maximal Slope in p(X).
3,401 citations
"Nonsmooth Differential Geometry: An..." refers background or result in this paper
...Notice that the next theorem is fully equivalent to the analogous statement proved for the Euclidean space in [8], the only difference being in the requirement that the curve has bounded compression, which has the effect of ‘averaging out the unsmoothness of the space’....
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...[8] and the references therein) ensures existence and uniqueness of a 1-parameter semigroup (ht)t≥0 of continuous operators from L 2(m) to itself such that for every f ∈ L2(m) the curve t 7→ ht(f) ∈ L2(m) is continuous on [0,∞), absolutely continuous on (0,∞) and fulfills...
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...completely characterize W2-absolutely continuous curves of measures provided one assumes that μt ≤ Cm for every t ∈ [0, 1] and some C > 0, in analogy with the result valid in the Euclidean space [8]....
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TL;DR: In this paper, the existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces were derived from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.
Abstract: We obtain some new existence, uniqueness and stability results for ordinary differential equations with coefficients in Sobolev spaces. These results are deduced from corresponding results on linear transport equations which are analyzed by the method of renormalized solutions.
2,075 citations
Book•
17 Mar 2000
TL;DR: In this paper, the authors introduce the concept of Finsler Manifolds and the fundamental properties of Minkowski Norms, and present an interesting family of examples of these properties.
Abstract: One Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 1.0 Physical Motivations.- 1.1 Finsler Structures: Definitions and Conventions.- 1.2 Two Basic Properties of Minkowski Norms.- 1.2 A. Euler's Theorem.- 1.2 B. A Fundamental Inequality.- 1.2 C. Interpretations of the Fundamental Inequality.- 1.3 Explicit Examples of Finsler Manifolds.- 1.3 A. Minkowski and Locally Minkowski Spaces.- 1.3 B. Riemannian Manifolds.- 1.3 C. Randers Spaces.- 1.3 D. Berwald Spaces.- 1.3 E. Finsler Spaces of Constant Flag Curvature.- 1.4 The Fundamental Tensor and the Cartan Tensor.- * References for Chapter 1.- 2 The Chern Connection.- 2.0 Prologue.- 2.1 The Vector Bundle ?*TM and Related Objects.- 2.2 Coordinate Bases Versus Special Orthonormal Bases.- 2.3 The Nonlinear Connection on the Manifold TM \0.- 2.4 The Chern Connection on ?*TM.- 2.5 Index Gymnastics.- 2.5 A. The Slash (...)s and the Semicolon (...) s.- 2.5 B. Covariant Derivatives of the Fundamental Tensor g.- 2.5 C. Covariant Derivatives of the Distinguished ?.- * References for Chapter 2.- 3 Curvature and Schur's Lemma.- 3.1 Conventions and the hh-, hv-, vv-curvatures.- 3.2 First Bianchi Identities from Torsion Freeness.- 3.3 Formulas for R and P in Natural Coordinates.- 3.4 First Bianchi Identities from "Almost" g-compatibility.- 3.4 A. Consequences from the $$ dx^k \wedge dx^l $$ Terms.- 3.4 B. Consequences from the $$ dx^k \wedge \frac{1} {F}\delta y^l $$ Terms.- 3.4 C. Consequences from the $$ \frac{1} {F}\delta y^k \wedge \frac{1} {F}\delta y^l $$ Terms.- 3.5 Second Bianchi Identities.- 3.6 Interchange Formulas or Ricci Identities.- 3.7 Lie Brackets among the $$ \frac{\delta } {{\delta x}} $$ and the $$ F\frac{\partial } {{\partial y}} $$.- 3.8 Derivatives of the Geodesic Spray Coefficients Gi.- 3.9 The Flag Curvature.- 3.9 A. Its Definition and Its Predecessor.- 3.9 B. An Interesting Family of Examples of Numata Type.- 3.10 Schur's Lemma.- *References for Chapter 3.- 4 Finsler Surfaces and a Generalized Gauss-Bonnet Theorem.- 4.0 Prologue.- 4.1 Minkowski Planes and a Useful Basis.- 4.1 A. Rund's Differential Equation and Its Consequence.- 4.1 B. A Criterion for Checking Strong Convexity.- 4.2 The Equivalence Problem for Minkowski Planes.- 4.3 The Berwald Frame and Our Geometrical Setup on SM.- 4.4 The Chern Connection and the Invariants I, J, K.- 4.5 The Riemannian Arc Length of the Indicatrix.- 4.6 A Gauss-Bonnet Theorem for Landsberg Surfaces.- *References for Chapter 4.- Two Calculus of Variations and Comparison Theorems.- 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature.- 5.1 The First Variation of Arc Length.- 5.2 The Second Variation of Arc Length.- 5.3 Geodesics and the Exponential Map.- 5.4 Jacobi Fields.- 5.5 How the Flag Curvature's Sign Influences Geodesic Rays.- *References for Chapter 5.- 6 The Gauss Lemma and the Hopf-Rinow Theorem.- 6.1 The Gauss Lemma.- 6.1 A. The Gauss Lemma Proper.- 6.1 B. An Alternative Form of the Lemma.- 6.1 C. Is the Exponential Map Ever a Local Isometry?.- 6.2 Finsler Manifolds and Metric Spaces.- 6.2 A. A Useful Technical Lemma.- 6.2 B. Forward Metric Balls and Metric Spheres.- 6.2 C. The Manifold Topology Versus the Metric Topology.- 6.2 D. Forward Cauchy Sequences, Forward Completeness.- 6.3 Short Geodesics Are Minimizing.- 6.4 The Smoothness of Distance Functions.- 6.4 A. On Minkowski Spaces.- 6.4 B. On Finsler Manifolds.- 6.5 Long Minimizing Geodesies.- 6.6 The Hopf-Rinow Theorem.- *References for Chapter 6.- 7 The Index Form and the Bonnet-Myers Theorem.- 7.1 Conjugate Points.- 7.2 The Index Form.- 7.3 What Happens in the Absence of Conjugate Points?.- 7.3 A. Geodesies Are Shortest Among "Nearby" Curves.- 7.3 B. A Basic Index Lemma.- 7.4 What Happens If Conjugate Points Are Present?.- 7.5 The Cut Point Versus the First Conjugate Point.- 7.6 Ricci Curvatures.- 7.6 A. The Ricci Scalar Ric and the Ricci Tensor Ricij.- 7.6 B. The Interplay between Ric and RiCij.- 7.7 The Bonnet-Myers Theorem.- *References for Chapter 7.- 8 The Cut and Conjugate Loci, and Synge's Theorem.- 8.1 Definitions.- 8.2 The Cut Point and the First Conjugate Point.- 8.3 Some Consequences of the Inverse Function Theorem.- 8.4 The Manner in Which cy and iy Depend on y.- 8.5 Generic Properties of the Cut Locus Cutx.- 8.6 Additional Properties of Cutx When M Is Compact.- 8.7 Shortest Geodesics within Homotopy Classes.- 8.8 Synge's Theorem.- *References for Chapter 8.- 9 The Cartan-Hadamard Theorem and Rauch's First Theorem.- 9.1 Estimating the Growth of Jacobi Fields.- 9.2 When Do Local Diffeomorphisms Become Covering Maps?.- 9.3 Some Consequences of the Covering Homotopy Theorem.- 9.4 The Cartan-Hadamard Theorem.- 9.5 Prelude to Rauch's Theorem.- 9.5 A. Transplanting Vector Fields.- 9.5 B. A Second Basic Property of the Index Form.- 9.5 C. Flag Curvature Versus Conjugate Points.- 9.6 Rauch's First Comparison Theorem.- 9.7 Jacobi Fields on Space Forms.- 9.8 Applications of Rauch's Theorem.- *References for Chapter 9.- Three Special Finsler Spaces over the Reals.- 10 Berwald Spaces and Szabo's Theorem for Berwald Surfaces.- 10.0 Prologue.- 10.1 Berwald Spaces.- 10.2 Various Characterizations of Berwald Spaces.- 10.3 Examples of Berwald Spaces.- 10.4 A Fact about Flat Linear Connections.- 10.5 Characterizing Locally Minkowski Spaces by Curvature.- 10.6 Szabo's Rigidity Theorem for Berwald Surfaces.- 10.6 A. The Theorem and Its Proof.- 10.6 B. Distinguishing between y-local and y-global.- *References for Chapter 10.- 11 Randers Spaces and an Elegant Theorem.- 11.0 The Importance of Randers Spaces.- 11.1 Randers Spaces, Positivity, and Strong Convexity.- 11.2 A Matrix Result and Its Consequences.- 11.3 The Geodesic Spray Coefficients of a Randers Metric.- 11.4 The Nonlinear Connection for Randers Spaces.- 11.5 A Useful and Elegant Theorem.- 11.6 The Construction of y-global Berwald Spaces.- 11.6 A. The Algorithm.- 11.6 B. An Explicit Example in Three Dimensions.- *References for Chapter 11 309.- 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem.- 12.0 Prologue.- 12.1 Characterizations of Constant Flag Curvature.- 12.2 Useful Interpretations of ? and E.- 12.3 Growth Rates of Solutions of E + ? E = 0.- 12.4 Akbar-Zadeh's Rigidity Theorem.- 12.5 Formulas for Machine Computations of K.- 12.5 A. The Geodesic Spray Coefficients.- 12.5 B. The Predecessor of the Flag Curvature.- 12.5 C. Maple Codes for the Gaussian Curvature.- 12.6 A Poincare Disc That Is Only Forward Complete.- 12.6 A. The Example and Its Yasuda-Shimada Pedigree.- 12.6 B. The Finsler Function and Its Gaussian Curvature.- 12.6 C. Geodesics Forward and Backward Metric Discs.- 12.6 D. Consistency with Akbar-Zadeh's Rigidity Theorem.- 12.7 Non-Riemannian Projectively Flat S2 with K = 1.- 12.7 A. Bryant's 2-parameter Family of Finsler Structures.- 12.7 B. A Specific Finsler Metric from That Family.- *References for Chapter 12 350.- 13 Riemannian Manifolds and Two of Hopf's Theorems.- 13.1 The Levi-Civita (Christoffel) Connection.- 13.2 Curvature.- 13.2 A. Symmetries, Bianchi Identities, the Ricci Identity.- 13.2 B. Sectional Curvature.- 13.2 C. Ricci Curvature and Einstein Metrics.- 13.3Warped Products and Riemannian Space Forms.- 13.3 A. One Special Class of Warped Products.- 13.3 B. Spheres and Spaces of Constant Curvature.- 13.3 C. Standard Models of Riemannian Space Forms.- 13.4 Hopf's Classification of Riemannian Space Forms.- 13.5 The Divergence Lemma and Hopf's Theorem.- 13.6 The Weitzenbock Formula and the Bochner Technique.- *References for Chapter 13.- 14 Minkowski Spaces, the Theorems of Deicke and Brickell.- 14.1 Generalities and Examples.- 14.2 The Riemannian Curvature of Each Minkowski Space.- 14.3 The Riemannian Laplacian in Spherical Coordinates.- 14.4 Deicke's Theorem.- 14.5 The Extrinsic Curvature of the Level Spheres of F.- 14.6 The Gauss Equations.- 14.7 The Blaschke-Santalo Inequality.- 14.8 The Legendre Transformation.- 14.9 A Mixed-Volume Inequality, and Brickell's Theorem.- * References for Chapter 14.
1,726 citations
"Nonsmooth Differential Geometry: An..." refers result in this paper
...This notion of gradient is in line with the one used in Finsler geometry, see for instance [19]....
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