• • • Riemannian geometry is the branch of that studies, with a Riemannian metric, i.e. With an on the at each point that varies from point to point. This gives, in particular, local notions of,,. From those, some other global quantities can be derived by local contributions. Riemannian geometry originated with the vision of expressed in his inaugural lecture ' ('On the Hypotheses on which Geometry is Based'). It is a very broad and abstract generalization of the in R 3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of on them, with techniques that can be applied to the study of of higher dimensions.

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The reader interested in Riemannian geometry itself is referred to [B-G-M] Chapters I and II, [C-E] or [CO], for more details and proofs. All manifolds we shall consider will be connected manifolds (unless otherwise stated). A Riemannian manifold (M,g) is a manifold M equipped with a Riemannian metric g: for any point x in M,gx is a scalar.

It enabled the formulation of 's, made profound impact on and, as well as, and spurred the development of. Riemannian geometry was first put forward in generality by in the 19th century. It deals with a broad range of geometries whose properties vary from point to point, including the standard types of. Any smooth manifold admits a, which often helps to solve problems of. It also serves as an entry level for the more complicated structure of, which (in four dimensions) are the main objects of the. Other generalizations of Riemannian geometry include.

There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. And produce torsions and curvature. The following articles provide some useful introductory material: • • • • • • • Classical theorems [ ] What follows is an incomplete list of the most classical theorems in Riemannian geometry. Coolscan 8000 ed drivers.

The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by and D. Ebin (see below). The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about. General theorems [ ] • The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ( M) where χ( M) denotes the of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see.

• also called. They state that every can be isometrically in a R n. Geometry in large [ ] In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at 'sufficiently large' distances. Pinched [ ] •. If M is a simply connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere.

• Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism) compact n-dimensional Riemannian manifolds with sectional curvature K ≤ C, diameter ≤ D and volume ≥ V. There is an ε n >0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature K ≤ ε n and diameter ≤ 1 then its finite cover is diffeomorphic to a.