Titlebook: Surveys in Geometry II; Athanase Papadopoulos Book 2024 The Editor(s) (if applicable) and The Author(s), under exclusive license to Spring

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,Double Forms, Curvature Integrals and the Gauss–Bonnet Formula,umulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for higher-dimensional Riemannian manifolds. It relates the Euler characteristic of a Riemannian manifold to a curvature integral over the manifold plus a somewhat enigmatic boundary term. In this chapter, we revisit t

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,Quaternions, Monge–Ampère Structures and ,-Surfaces,d widespread applications in hyperbolic geometry, general relativity, Teichmüller theory, and so on. In this chapter, we present a quaternionic reformulation of these ideas. This yields simpler proofs of the main results whilst pointing towards the higher-dimensional generalisation studied by the au

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Lagrangian Grassmannians of Polarizations,ures on a real vector space, consisting of an inner product, a symplectic form, and a complex structure. A polarization is a decomposition of the complexified vector space into the eigenspaces of the complex structure; this information is equivalent to the specification of a compatible triple. When

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On the Geometry of Finite Homogeneous Subsets of Euclidean Spaces,of regular and semiregular polytopes in Euclidean spaces by whether or not their vertex sets have the normal homogeneity property or the Clifford–Wolf homogeneity property. Every finite homogeneous metric subspace of a Euclidean space represents the vertex set of a compact convex polytope whose isom

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Lagrangian Grassmannians of Polarizations, This introduction would be useful for those interested in applications of polarizations to representation theory, loop groups, complex geometry, moduli spaces, quantization, and conformal field theory.