* Curvature vector and geodesic curvature for appropriate notions of curvature of curves in Riemannian manifolds, of any dimension

* Curvature of Riemannian manifolds

* Curvature of Riemannian manifolds

An important question related to Curvature invariants is when the set of polynomial curvature invariants can be used to ( locally ) distinguish manifolds.

* Curvature form for the appropriate notion of curvature for vector bundles and principal bundles with connection

* Curvature of a measure for a notion of curvature in measure theory

Curvature of the earth limits the reach of the signal, but due to the high orbit of the satellites, two or three are usually sufficient to provide coverage for an entire continent.

* D. Rawlins: " Methods for Measuring the Earth's Size by Determining the Curvature of the Sea " and " Racking the Stade for Eratosthenes ", appendices to " The Eratosthenes-Strabo Nile Map.

* Module for Curvature

Kerr also founded Curvature Wines, which helps to raise money for breast cancer charities.

* Curvature invariant, for curvature invariants in a more general context.

:* Curvature myopia is attributed to excessive, or increased, curvature of one or more of the refractive surfaces of the eye, especially the cornea.

; Curvature of Field: The Petzval field curvature means that the image instead of lying in a plane actually lies on a curved surface which is described as hollow or round.

Curvature invariants in general relativity are a set of scalars called curvature invariants that arise in general relativity.

For instance, volume and Riemannian curvature are invariants that can distinguish different geometric structures on the same smooth manifold — that is, one can smoothly " flatten out " certain manifolds, but it might require distorting the space and affecting the curvature or volume.

Thus differential geometry may study differentiable manifolds equipped with a connection, a metric ( which may be Riemannian, pseudo-Riemannian, or Finsler ), a special sort of distribution ( such as a CR structure ), and so on.

Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric.

Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds.

A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry.

An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.

Bombieri is also known for his pro bono service on behalf of the mathematics profession, e. g. for serving on external review boards and for peer-reviewing extraordinarily complicated manuscripts ( like the papers of John Nash on embedding Riemannian manifolds and of Per Enflo on the invariant subspace problem ).

A diffeomorphism between two Riemannian manifolds is called a conformal map if the pulled back metric is conformally equivalent to the original one.

Let ( M, g ) and ( N, h ) be Riemannian manifolds.

Analogously, isometric immersion is an immersion between Riemannian manifolds which preserves the Riemannian metrics.

To extend de Broglie – Bohm theory to curved space ( Riemannian manifolds in mathematical parlance ), one simply notes that all of the elements of these equations make sense, such as gradients and Laplacians.

Bernhard Riemann, in a famous lecture in 1854, founded the field of Riemannian geometry, discussing in particular the ideas now called manifolds, Riemannian metric, and curvature.

Hamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknown simply connected closed 3-manifold.

The later invention of non-Euclidean geometry does not resolve this question ; for one might as well ask, " If given the axioms of Riemannian geometry, can an omnipotent being create a triangle whose angles do not add up to more than 180 degrees?

In Riemannian geometry, two Riemannian metrics and on smooth manifold are called conformally equivalent if for some positive function on.

For example, it follows that any closed oriented Riemannian surface can be C < sup > 1 </ sup > isometrically embedded into an arbitrarily small ε-ball in Euclidean 3-space ( there is no such C < sup > 2 </ sup >- embedding since from the formula for the Gauss curvature an extremal point of such an embedding would have curvature ≥ ε < sup >- 2 </ sup >).

The technical statement is as follows: if M is a given m-dimensional Riemannian manifold ( analytic or of class C < sup > k </ sup >, 3 ≤ k ≤ ∞), then there exists a number n ( with n ≤ m ( 3m + 11 )/ 2 if M is a compact manifold, or n ≤ m ( m + 1 )( 3m + 11 )/ 2 if M is a non-compact manifold ) and an injective map ƒ: M → R < sup > n </ sup > ( also analytic or of class C < sup > k </ sup >) such that for every point p of M, the derivative dƒ < sub > p </ sub > is a linear map from the tangent space T < sub > p </ sub > M to R < sup > n </ sup > which is compatible with the given inner product on T < sub > p </ sub > M and the standard dot product of R < sup > n </ sup > in the following sense:

He constructed an infinite family of geometries which are not Euclidean by giving a formula for a family of Riemannian metrics on the unit ball in Euclidean space.

There is a key distinction between extrinsic curvature, which is defined for objects embedded in another space ( usually a Euclidean space ) in a way that relates to the radius of curvature of circles that touch the object, and intrinsic curvature, which is defined at each point in a Riemannian manifold.

This generalizes the notion of geodesic for Riemannian manifolds.

Conversely, except in dimension 2, if the curvature of a Riemannian manifold has this form for some function K, then the Bianchi identities imply that K is constant and thus that the manifold is ( locally ) a space form.

Gauss's contributions to non-Euclidean geometry laid the groundwork for the subsequent development of Riemannian geometry by Bernhard Riemann.

If q < sub > m </ sub > is positive for all non-zero X < sub > m </ sub >, then the metric is positive definite at m. If the metric is positive definite at every m ∈ M, then g is called a Riemannian metric.

In the theory of Riemannian and pseudo-Riemannian manifolds the term covariant derivative is often used for the Levi-Civita connection.

In Riemannian geometry for example, ramification is a generalization of the notion of covering maps.

In geometry, methods of ergodic theory have been used to study the geodesic flow on Riemannian manifolds, starting with the results of Eberhard Hopf for Riemann surfaces of negative curvature.

Two important special cases of this are the exponential map for a manifold with a Riemannian metric, and the exponential map from a Lie algebra to a Lie group.

Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while ( for dimension at least 3 ), negative Ricci curvature has no topological implications.

# The Cheeger-Gromoll splitting theorem states that if a complete Riemannian manifold with contains a line, meaning a geodesic γ such that for all, then it is isometric to a product space.

In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only pointwise assertion ; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.