By Marcel Berger
In the course of its first hundred years, Riemannian geometry loved regular, yet undistinguished progress as a box of arithmetic. within the final fifty years of the 20 th century, notwithstanding, it has exploded with task. Berger marks the beginning of this era with Rauch's pioneering paper of 1951, which incorporates the 1st actual pinching theorem and an grand bounce within the intensity of the relationship among geometry and topology. considering that then, the sphere has turn into so wealthy that it's nearly very unlikely for the uninitiated to discover their method via it. Textbooks at the topic constantly needs to pick out a selected technique, hence narrowing the path.In this ebook, Berger presents a very extraordinary survey of the most advancements in Riemannian geometry within the final fifty years. probably the most strong gains of Riemannian manifolds is they have invariants of (at least) 3 other forms. There are the geometric invariants: topology, the metric, numerous notions of curvature, and relationships between those. There are analytic invariants: eigenvalues of the Laplacian, wave equations, Schrodinger equations. There are the invariants that come from Hamiltonian mechanics: geodesic stream, ergodic homes, periodic geodesics.Finally, there are very important effects bearing on sorts of invariants. to maintain the scale of this survey possible, Berger makes a speciality of 5 parts of Riemannian geometry: Curvature and topology; the development of and the type of house varieties; distinct metrics, particularly Einstein metrics; eigenvalues and eigenfunctions of the Laplacian; the research of periodic geodesics and the geodesic movement. different themes are taken care of in much less element in a separate section.While Berger's survey isn't meant for the total newbie (one may still already be accustomed to notions of curvature and geodesics), he offers a close map to the foremost advancements of Riemannian geometry from 1950 to 1999. very important threads are highlighted, with short descriptions of the consequences that make up that thread. This supremely scholarly account is impressive for its cautious citations and voluminous bibliography. if you want to find out about the consequences that experience outlined Riemannian geometry within the final part century, commence with this ebook.
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