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Basic Results of Integral Geometry
The results in this chapter are of a geometric nature, and they emanate from the basic concept of motion-invariant measure associated with a mobile...
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Integral Geometry of Pairs of Lines and Planes
In this paper, we present some results obtained previously in Cufí et al. (J. Math. Anal. Appl. 458(1):436–451 (2018); Mathematika 65(4):874–896...
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Problems of Extrinsic Geometry of Foliations
This survey is devoted to particular problems of extrinsic geometry of foliations, which, roughly speaking, describes how leaves (or single...
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Comparison Geometry for Integral Radial Bakry-Émery Ricci Tensor Bounds
In this paper we prove mean curvature comparisons and volume comparisons on a smooth metric measure space when the integral radial Bakry-Émery Ricci...
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Dirac Geometry I: Commutative Algebra
The purpose of this paper and its sequel is to develop the geometry built from the commutative algebras that naturally appear as the homology of...
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A Dual Yamabe Flow and Related Integral Flows
The author studies a family of nonlinear integral flows that involve Riesz potentials on Riemannian manifolds. In the Hardy-Littlewood-Sobolev (HLS...
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Geometry of Linear Algebras
In this paper, we consider spaces whose geometry is generated by a homogeneous function of degree m ≥ 2, which is invariant under the action of some...
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Vector Calculus and Geometry
Vector calculus is a generalisation of calculus concepts (i.e. differentiation and integration) from...
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Integral Formulas and Conformal Vector Fields
This chapter is devoted to the description of some integration theorems, integral formulas, operators and important results on conformal vector...
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Legendre duality: from thermodynamics to information geometry
This paper reviews the role of convex duality in Information Geometry. It clarifies the notion of bi-orthogonal coordinates associated with Legendre...
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Hypercomplex Cauchy-Type Integral
In this chapter, we consider a certain hypercomplex Cauchy-type integral taking values in a three-dimensional harmonic algebra with two-dimensional...
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Geometry of Fibered Graphs of Map**s
In this paper, we examine the differential-geometric aspect of constant-rank map**s of smooth manifolds based on the concept of a graph as a smooth...
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Integral Representations and Integral Identities
Working in the complex plane, in §1.1 we prove versions of the classical Cauchy-Pompeiu integral representation formula, Morera’s Theorem, the...
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Introduction to Logarithmic Geometry
These chapter is based on a minicourse on logarithmic resolution of singularities given by the author, and it provides an extended version of its...
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The Intrinsic Geometry of Surfaces
By intrinsic geometry of a surface we mean those properties that depend exclusively on measurements made on the surface but not on how the surface is...
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The Geometry of the Gauss Map
In this chapter we deal with the extrinsic geometry of the surface, by defining quantities (curvatures) that express how the surface is embedded in...
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Elements of Algebraic Geometry
Here we present the algebraic geometry background for the study of PAC fields. The central result is a descent argument which associates to each...
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Classifications of Dupin hypersurfaces in Lie sphere geometry
This is a survey of local and global classification results concerning Dupin hypersurfaces in S n (or R n ) that have been obtained in the context of...
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Affine convex geometry – Part 2
The main objective of this chapter is to study some important (starlike or convex) bodies associated to a given convex body. These are the so-called...