Cover

Title page

Introduction

Aim and key ideas

Overview of the content

Some open problems

Chapter 1. The machinery of 𝐿^{𝑝}(\mm)-normed modules

1.1. Assumptions and notation

1.2. Basic definitions and properties

1.3. Alteration of the integrability

1.4. Local dimension

1.5. Tensor and exterior products of Hilbert modules

1.6. Pullback

Chapter 2. First order differential structure of general metric measure spaces

2.1. Preliminaries: Sobolev functions on metric measure spaces

2.2. Cotangent module

2.2.1. The construction

2.2.2. Differential of a Sobolev function

2.3. Tangent module

2.3.1. Tangent vector fields and derivations

2.3.2. On the duality between differentials and gradients

2.3.3. Divergence

2.3.4. Infinitesimally Hilbertian spaces

2.3.5. In which sense the norm on the tangent space induces the distance

2.4. Maps of bounded deformation

2.5. Some comments

Chapter 3. Second order differential structureof \RCD(𝐾,∞) spaces

3.1. Preliminaries: \RCD(𝐾,∞) spaces

3.2. Test objects and some notation

3.3. Hessian

3.3.1. The Sobolev space 𝑊^{2,2}(\X)

3.3.2. Why there are many 𝑊^{2,2} functions

3.3.3. Calculus rules

3.3.3.1. Some auxiliary Sobolev spaces

3.3.3.2. Statement and proofs of calculus rules

3.4. Covariant derivative

3.4.1. The Sobolev space 𝑊^{1,2}_{𝐶}(𝑇\X)

3.4.2. Calculus rules

3.4.3. Second order differentiation formula

3.4.4. Connection Laplacian and heat flow of vector fields

3.5. Exterior derivative

3.5.1. The Sobolev space 𝑊^{1,2}_{}(Λ^{𝑘}𝑇*\X)

3.5.2. de Rham cohomology and Hodge theorem

3.6. Ricci curvature

Bibliography

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