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Titlebook: Aspects of Differential Geometry II; Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenz Book 2015 Springer Nature Switzerland AG 2015

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期刊全称Aspects of Differential Geometry II
影响因子2023Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenz
视频video
学科分类Synthesis Lectures on Mathematics & Statistics
图书封面Titlebook: Aspects of Differential Geometry II;  Peter Gilkey,JeongHyeong Park,Ramón Vázquez-Lorenz Book 2015 Springer Nature Switzerland AG 2015
影响因子Differential Geometry is a wide field. We have chosen to concentrate upon certain aspects that are appropriate for an introduction to the subject; we have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are studied, and the volume of geodesic balls is treated. An introduction to both holomorphic and Kähler geometry is given. In Chapter 5, the basic properties of de Rham cohomology are discussed, the Hodge Decomposition Theorem, Poincaré duality, and the Künneth formula are proved, and a brief introduction to the theory of characteristic classes is given. In Chapter 6, Lie groups and Lie algebras are dealt with. The exponential map, the classical groups, and geodesics in the context of a bi-invariant metric are discussed. The de Rham cohomology ofcompact Lie groups and the Peter--Weyl Theorem are treated. In Chapter 7, material concerning homogeneous spaces and symmetric spaces is presented. Book II concludes in Chapter 8 where the relationship
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Homogeneous Spaces and Symmetric Spaces,ion 7.3, we treat Killing vector fields. In Section 7.4, we establish facts concerning homogeneous pseudo-Riemannian manifolds. In Section 7.5, we use Jacobi vector fields to study the geometry of local symmetric spaces. In Section 7.6, we examine symmetric spaces.
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Other Cohomology Theories, de Rham cohomology and by showing that de Rham cohomology is a homotopy functor. In Section 8.2, we discuss simplicial theory. In Section 8.3, we discuss singular (i.e., topological or simply TP) cohomology . and .. In Section 8.4, we treat sheaf cohomology.
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Book 2015have not attempted an encyclopedic treatment. Book II deals with more advanced material than Book I and is aimed at the graduate level. Chapter 4 deals with additional topics in Riemannian geometry. Properties of real analytic curves given by a single ODE and of surfaces given by a pair of ODEs are
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Additional Topics in Riemannian Geometry,Section 4.3, we turn to holomorphic geometry. We present the Cauchy-Riemann equations, define almost complex structures, and state the Newlander-Nirenberg Theorem. We decompose . and introduce the spaces .. We define complex projective space. In Section 4.4, we present an introduction to Kähler geometry.
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Homogeneous Spaces and Symmetric Spaces, 7.1, we prove results about smooth structures on coset spaces. In Section 7.2, we discuss the isometry group of a pseudo-Riemannian manifold. In Section 7.3, we treat Killing vector fields. In Section 7.4, we establish facts concerning homogeneous pseudo-Riemannian manifolds. In Section 7.5, we use
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