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In Differential Geometry , Riemannian geometry is the study of Smooth Manifolds with Riemannian Metric s, i.e. a choice of Positive-definite Quadratic Form on a Manifold 's Tangent Space s which varies smoothly from point to point. This gives in particular local ideas of Angle , Length Of Curves , and Volume . From those some other global quantities can be derived by Integrating local contributions. INTRODUCTION Riemannian geometry was first put forward in generality by Bernhard Riemann in the Nineteenth Century . It deals with a broad range of geometries whose metric properties vary from point to point, as well as two standard types of Non-Euclidean Geometry , Spherical Geometry and Hyperbolic Geometry , as well as Euclidean Geometry itself. Any smooth manifold admits a Riemannian Metric , which often helps to solve problems of Differential Topology . It also serves as an entry level for the more complicated structure of Pseudo-Riemannian Manifold s, which (in four dimensions) are the main objects of the Theory Of General Relativity . There is no easy introduction to Riemannian geometry. It is generally recommended that one should work in the subject for quite a while to build some geometric intuition, usually by doing enormous amounts of calculations. The following articles might serve as a rough introduction: # Metric Tensor # Riemannian Manifold # Levi-Civita Connection # Curvature # Curvature Tensor . The following articles might be also useful: # List Of Differential Geometry Topics # Glossary Of Riemannian And Metric Geometry CLASSICAL THEOREMS IN RIEMANNIAN GEOMETRY What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance, beauty, and simplicity of formulation. 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 # Gauss-Bonnet Theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to where denotes the Euler Characteristic of ''M''. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see Generalized Gauss-Bonnet Theorem . # Nash Embedding Theorem s also called Fundamental Theorems Of Riemannian Geometry . They state that every Riemannian Manifold can be isometrically Embedded in a Euclidean Space '''R'''''n''. Local to global theorems 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 Sectional Curvature #1/4-pinched Sphere Theorem. If ''M'' is a complete ''n''-dimensional Riemannian manifold with sectional curvature strictly pinched between 1 and 4 then ''M'' is homeomorphic to ''n''-sphere. | ||
| #''' | "http://wwwinformationdelightinfo/encyclopedia/entry/Almost_flat_manifold" class="copylinks">Gromov's Almost Flat Manifolds ''' There is an <math>\epsilon_n>0</math> such that if an ''n''-dimensional Riemannian manifold has a metric with sectional curvature <math>K\le \epsilon_n</math> and diameter <math>\le 1</math> then its finite cover is diffeomorphic to a Nil Manifold |
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