Differential Geometry

Read e-book online Riemannian Geometry and Geometric Analysis PDF

By Jürgen Jost

ISBN-10: 3540259074

ISBN-13: 9783540259077

This confirmed reference paintings maintains to steer its readers to a few of the most well liked issues of latest mathematical learn. in addition to a number of smaller additions, reorganizations, corrections, and a scientific bibliography, the most new gains of the 4th version are a scientific creation to Kähler geometry and the presentation of extra recommendations from geometric analysis.

From the stories: "This ebook presents a truly readable advent to Riemannian geometry and geometric research. the writer specializes in utilizing analytic equipment within the examine of a few basic theorems in Riemannian geometry, e.g., the Hodge theorem, the Rauch comparability theorem, the Lyusternik and Fet theorem and the lifestyles of harmonic mappings. With the massive improvement of the mathematical topic of geometric research, the current textbook is such a lot welcome. [..] The ebook is made extra attention-grabbing through the views in a variety of sections." Math. Reviews

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2 Let G be a subgroup of Gl(n, R), for example O(n) or SO(n), the orthogonal or special orthogonal group. We say that a vector bundle has the structure group G if there exists an atlas of bundle charts for which all transition maps have their values in G. 3 Let (E, π, M ) be a vector bundle. A section of E is a differentiable map s : M → E with π ◦ s = idM . The space of sections of E is denoted by Γ (E). We have already seen an example of a vector bundle above, namely the tangent bundle T M of a differentiable manifold M.

T. this basis, we obtain a map Φ : Tp M → Rd v = v i ei → (v 1 , . . , v d ). For the subsequent construction, we identify Tp M with Rd via Φ. 3, there exists a neighborhood U of p which is mapped by exp−1 p diffeomorphically onto a neighborhood of 0 ∈ Tp M, hence, with our identification Tp M ∼ = Rd , diffeomorphically onto a neighborhood Ω of 0 ∈ Rd . In particular, p is mapped to 0. 4 The local coordinates defined by the chart (exp−1 p , U ) are called (Riemannian) normal coordinates with center p.

3). In our presentation, we only consider finite dimensional Riemannian manifolds. It is also possible, and often very useful, to introduce infinite dimensional Riemannian manifolds. Those are locally modeled on Hilbert spaces instead of Euclidean ones. The lack of local compactness leads to certain technical complications, but most ideas and constructions of Riemannian geometry pertain to the infinite dimensional case. Such infinite dimensional manifolds arise for example naturally as certain spaces of curves on finite dimensional Riemannian manifolds.

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Riemannian Geometry and Geometric Analysis by Jürgen Jost

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