differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry...

In mathematics, curvature is any of several strongly related concepts in geometry that intuitively measure the amount by which a curve deviates from being...

the manifold a symmetric bilinear form (Besse 1987, p. 43). Broadly, one could analogize the role of the Ricci curvature in Riemannian geometry to that of...

introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications...

The main tensorial invariant of a connection form is its curvature form. In the presence of a solder form identifying the vector bundle with the tangent...

(thoracic and sacral curvatures) form during fetal development. The secondary curves develop after birth. The cervical curvature forms as a result of lifting...

f12 = Ez/c, f23 = −Bz, or equivalent definitions. This form is a special case of the curvature form on the U(1) principal bundle on which both electromagnetism...

equal to the integral of a certain polynomial (the Euler class) of its curvature form (an analytical invariant). It is a highly non-trivial generalization...

term curvature tensor may refer to: the Riemann curvature tensor of a Riemannian manifold — see also Curvature of Riemannian manifolds; the curvature of...

metric whose curvature form ω is positive (since ω is then a Kähler form that represents the first Chern class of L in H2(X, Z)). The Kähler form ω that satisfies...

Gaussian curvature or Gauss curvature Κ of a smooth surface in three-dimensional space at a point is the product of the principal curvatures, κ1 and κ2...

Z\right)+R\left(T\left(X,Y\right),Z\right)\right)=0} The curvature form is the gl(n)-valued 2-form Ω = D ω = d ω + ω ∧ ω {\displaystyle \Omega =D\omega =d\omega...

geometry, the two principal curvatures at a given point of a surface are the maximum and minimum values of the curvature as expressed by the eigenvalues...

In mathematics, the mean curvature H {\displaystyle H} of a surface S {\displaystyle S} is an extrinsic measure of curvature that comes from differential...

fundamental form Gauss–Codazzi–Mainardi equations Dupin indicatrix Asymptotic curve Curvature Principal curvatures Mean curvature Gauss curvature Elliptic...

Spherical Earth or Earth's curvature refers to the approximation of the figure of the Earth as a sphere. The earliest documented mention of the concept...

\mathbf {Q} )} can be presented as differential forms which depend polynomially on the curvature form of a vector bundle. This Chern–Weil theory revealed...

{\displaystyle P\times ^{G}W} . The curvature form of a principal G-connection ω is the g {\displaystyle {\mathfrak {g}}} -valued 2-form Ω defined by Ω = d ω + 1...

formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor. Consider the following problem...

best approximates the curvature of a curve at a specific point. It is tangent to the curve at that point and has the same curvature as the curve at that...

geometry, the sectional curvature is one of the ways to describe the curvature of Riemannian manifolds. The sectional curvature K(σp) depends on a two-dimensional...

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno...

in terms of connections and curvature representing classes in the de Rham cohomology rings of M. That is, the theory forms a bridge between the areas of...

nondegenerate bilinear form over g {\displaystyle {\mathfrak {g}}} (if G is semisimple, the Killing form will do) and F is the curvature form F ≡ d A + A ∧ A...

one can write down concretely any k-form of the closed connection ω as some 2k-form of the associated curvature form Ω of ω. In 1974 S. S. Chern and J....

mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a...

and measure the Earth's radius involve either the spheroid's radius of curvature or the actual topography. A few definitions yield values outside the range...

fundamental form, it serves to define extrinsic invariants of the surface, its principal curvatures. More generally, such a quadratic form is defined for...

}\wedge \sigma .} A flat connection is one whose curvature form vanishes identically. The curvature form has a local description called Cartan's structure...

classes can be expressed as polynomials in the coefficients of the curvature form. There are various ways of approaching the subject, each of which focuses...