## Riemannian geometry

In mathematics, a smooth compact manifold M is called almost flat if for any $\epsilon>0$ there is a Riemannian metric $g_\epsilon$ on M such that $\mbox\left\{diam\right\}\left(M,g_\epsilon\right)\le 1$ and ...more on Wikipedia about "Almost flat manifold"

In mathematics, the Bishop–Gromov inequality is a classical theorem in Riemannian geometry. It is the key point in the proof of Gromov's compactness theorem. ...more on Wikipedia about "Bishop–Gromov inequality"

In differential geometry, the Calabi flow is a process which deforms the metric of a Riemannian manifold (or better yet, a Kähler manifold) in a manner formally analogous to the way that vibrations are damped and dissipated in a hypothetical curved n-dimensional structural element. ...more on Wikipedia about "Calabi flow"

Cartan's method was adapted and improved for general relativity by A. Karlhede, who gave the first algorithmic description of what is now called the Cartan-Karlhede algorithm. The algorithm was soon implemented by J. Åman in an early symbolic computation engine, SHEEP (symbolic computation system), but the size of the computations proved too challenging for early computer systems to handle. ...more on Wikipedia about "Cartan-Karlhede algorithm"

In mathematics and physics, the Christoffel symbols, named for Elwin Bruno Christoffel ( 1829– 1900), are coordinate-space expressions for the Levi-Civita connection derived from the metric tensor. The Christoffel symbols are used whenever practical calculations involving geometry must be performed, as they allow very complex calculations to be performed without confusion. Unfortunately, they are usually quite lengthy, and require careful attention to detail. By contrast, the index-less, formal notation for the Levi-Civita connection is terse, and allows theorems to be stated in an elegant way, but is next to useless for practical calculations. ...more on Wikipedia about "Christoffel symbols"

In mathematics, a conformal map is a function which preserves angles. ...more on Wikipedia about "Conformal map"

In mathematics, constant curvature in differential geometry is a concept most commonly applied to surfaces. For those the scalar curvature is a single number determining the local geometry, and its constancy has the obvious meaning that it is the same at all points. The circle has constant curvature, also, in a natural (but different) sense. ...more on Wikipedia about "Constant curvature" Must see http://www.shortopedia.com

In mathematics, constraint counting is a crude but often useful way of counting the number of free functions needed to specify a solution to a partial differential equation. ...more on Wikipedia about "Constraint counting"

In differential geometry, the Cotton tensor on a (pseudo)- Riemannian manifold of dimension n is a third-order tensor concomitant of the metric, like the Weyl tensor. Just as the vanishing of the Weyl tensor for n ≥ 4 is a necessary and sufficient condition for the manifold to be conformally flat, the same is true for the Cotton tensor for n = 3, while for n < 3 it is identically zero. ...more on Wikipedia about "Cotton tensor"

In mathematics and theoretical physics, covariance and contravariance are concepts used in many areas, generalising in a sense invariance, i.e. the property of being unchanged under some transformation. In mathematical terms they occur in a foundational way in linear algebra and multilinear algebra, differential geometry and other branches of geometry, category theory and algebraic topology. In physics they are important to the treatment of vectors and other quantities, such as tensors, that have physical meaning but are not scalars. Both special relativity ( Lorentz covariance) and general relativity ( general covariance) use covariant basis vectors. ...more on Wikipedia about "Covariance and contravariance"

In mathematics, the covariant derivative is a way of specifying a derivative along vector fields on a manifold. ...more on Wikipedia about "Covariant derivative"

Curvature is the amount by which a geometric object deviates from being flat. ...more on Wikipedia about "Curvature"

The linear transformation $w\mapsto R\left(u,v\right)w$ is also called the curvature transformation or endomorphism. ...more on Wikipedia about "Curvature of Riemannian manifolds"

In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. The curvature tensor is given in terms of a Levi-Civita connection (more generally, an affine connection) $\nabla$(or covariant differentiation) by the following formula: ...more on Wikipedia about "Curvature tensor" www.shortopedia.com, just the best.

In mathematics and physics, n-dimensional de Sitter space, denoted $dS_n$, is the maximally symmetric, simply-connected, Lorentzian manifold with constant positive curvature. It may be regarded as the Lorentzian analog of an n-sphere (with its canonical Riemannian metric). ...more on Wikipedia about "De Sitter space"

An Einstein manifold is a Riemannian manifold (M,g) whose Ricci tensor is proportional to the metric tensor: ...more on Wikipedia about "Einstein manifold"

:For other topics related to Einstein, see Einstein (disambiguation). ...more on Wikipedia about "Einstein notation"

There are two different (but closely related) notions of an exponential map in differential geometry, both of which generalize the ordinary exponential function of mathematical analysis. ...more on Wikipedia about "Exponential map"

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that given a Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free connection preserving the metric tensor. ...more on Wikipedia about "Fundamental theorem of Riemannian geometry"

In differential geometry, the Gauss map (named, like so many things, after Carl F. Gauss) maps a surface in Euclidean space R3 to the unit sphere S2. Namely, given a surface X lying in R3, the Gauss map is a continuous map N: XS2 such that N(p) is orthogonal to X at p. ...more on Wikipedia about "Gauss map"

In mathematics, a geodesic is a generalization of the notion of a "straight line" to "curved spaces". ...more on Wikipedia about "Geodesic" The article you are reading is from shortopedia Riemannian_geometry

In differential geometry, the geodesic curvature vector is a property of curves in a metric space which reflects the deviance of the curve from following the shortest arc length distance along each infinitesimal segment of its length. ...more on Wikipedia about "Geodesic curvature"

In differential geometry, the geodesic deviation equation is an equation involving the Riemann curvature tensor, which measures the change in separation of neighbouring ...more on Wikipedia about "Geodesic deviation equation"

The geometrization conjecture, also known as Thurston's geometrization conjecture, concerns the geometric structure of compact 3-manifolds. It was proposed by William Thurston in the late 1970s. It 'includes' other conjectures, such as the Poincaré conjecture and the Thurston elliptization conjecture. Here are some essential concepts used in the conjecture: ...more on Wikipedia about "Geometrization conjecture"

Alexandrov space a generalization of Riemannian manifolds with upper, lower or integral curvature bounds (the last one works only in dimension 2) ...more on Wikipedia about "Glossary of Riemannian and metric geometry"