During this time he learned tensor calculus, a mathematical instrument invented by Gregorio Ricci and Tullio Levi-Civita, and needed to demonstrate the principles of general relativity.

Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely.

On the left-hand side is the Einstein tensor, a specific divergence-free combination of the Ricci tensor and the metric.

The Ricci tensor itself is related to the more general Riemann curvature tensor as

Here, R < sub > ij </ sub > is the Ricci tensor.

The definition of a tensor as a multidimensional array satisfying a transformation law traces back to the work of Ricci.

where is the Ricci tensor, is the Ricci scalar ( the tensor contraction of the Ricci tensor ), and is the universal gravitational constant.

This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor.

It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

The Ricci tensor is defined on any pseudo-Riemannian manifold, as a trace of the Riemann curvature tensor.

Like the metric itself, the Ricci tensor is a symmetric bilinear form on the tangent space of the manifold.

In relativity theory, the Ricci tensor is the part of the curvature of space-time that determines the degree to which matter will tend to converge or diverge in time ( via the Raychaudhuri equation ).

The Latinized name " Confucius " is derived from " Kong Fuzi ", which was first coined by 16th-century Jesuit missionaries to China, most probably by Matteo Ricci.

According to Carla Ricci, “ The place she Magdalene occupied in the list cannot be considered fortuitous ,” because over and over Mary Magdalene ’ s name is placed at the head of specifically named women, indicating her importance.

The metric is improved using the Ricci flow equations ;

where g is the metric and R its Ricci curvature,

This procedure is known as Ricci flow with surgery.

A special case of Perelman's theorems about Ricci flow with surgery is given as follows.

The first step is to deform the manifold using the Ricci flow.

The formula for the Ricci flow is an imitation of the heat equation which describes the way heat flows in a solid.

Perelman proved this note goes up as the manifold is deformed by the Ricci flow.

** Jesuit Matteo Ricci is allowed to enter the country.

* June 29 – Catherine of Ricci ( b. 1522 ) is canonized.

The difference in area of a sector of the disc is measured by the Ricci curvature.

The Ricci flow was defined by Richard Hamilton as a way to deform manifolds.

The Ricci flow was only defined for smooth differentiable manifolds.

Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions.

Let denote the tangent space of M at a point p. For any pair of tangent vectors at p, the Ricci tensor evaluated at is defined to be the trace of the linear map given by

The scalar curvature is defined as the trace of the Ricci tensor, and it can be characterized as a multiple of the average of the sectional curvatures at a point.

It is defined as the trace of the Ricci curvature tensor with respect to the metric:

Unlike the Riemann curvature tensor or the Ricci tensor, which both can be naturally defined for any affine connection, the scalar curvature requires a metric of some kind.

More generally, the Ricci tensor can be defined in broader class of metric geometries ( by means of the direct geometric interpretation, below ) that includes Finsler geometry.

If we consider the metric tensor ( and the associated Ricci tensor ) to be functions of a variable which is usually called " time " ( but which may have nothing to do with any physical time ), then the Ricci flow may be defined by the geometric evolution equation

The factor of − 2 is of little significance, since it can be changed to any nonzero real number by rescaling t. However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times ; if the sign is changed then the Ricci flow would usually only be defined for small negative times.

The Ricci tensor is defined as the contraction of the Riemann tensor.

The Ricci scalar is defined as

The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor.

That is, quaternion-Kähler is defined as one with holonomy reduced to Sp ( n )· Sp ( 1 ) and with non-zero Ricci curvature ( which is constant ).

They are defined in terms of the Weyl tensor and its left ( or right ) dual, the Ricci tensor, and the trace-free Ricci tensor

with R < sub > AB </ sub > the five-dimensional Ricci curvature, may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations

In Luke, the author writes that Jesus “ took Peter, John and James .” According to Ricci, because Peter occupies the first position in the list, that place can be considered the position of highest importance.

Machiavelli's literary executor, Giuliano de ' Ricci, also reported having seen that Machiavelli, his grandfather, made a comedy in the style of Aristophanes which included living Florentines as characters, and to be titled Le Maschere.

Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times.

The work of his late maturity can broadly be characterized as " neo-Riccian ", for in this period his style echoed in some respects his youthful emulation of Marco Ricci.

Although Fernão Pires de Andrade and his Portuguese comrades were the first to open up China to the West, another significant diplomatic mission reaching all the way to Beijing would not be carried out until an Italian, the Jesuit Matteo Ricci ( 1552 – 1610 ) ventured there in 1598.

As a Catholic missionary, Ricci strongly criticized the " recondite science " of geomancy along with astrology as yet another superstitio absurdissima of the heathens: " What could be more absurd than their imagining that the safety of a family, honors, and their entire existence must depend upon such trifles as a door being opened from one side or another, as rain falling into a courtyard from the right or from the left, a window opened here or there, or one roof being higher than another?

In 2006 Ricci stated that she feels that at 5 ft 1 in ( 155 cm ) she is " too short " to ever be an A-list actress, saying she tends " to look really small on camera ".

Having just served five years in prison for a crime he did not commit, Billy Brown ( Vincent Gallo ) kidnaps a young tap dancer named Layla ( Christina Ricci ) and forces her to pretend to be his wife.

This geometry can be modeled as a left invariant metric on the Bianchi group of type V. Under Ricci flow manifolds with hyperbolic geometry expand.

The case of 3-manifolds that should be spherical has been slower, but provided the spark needed for Richard Hamilton to develop his Ricci flow.

The idea is that the Ricci flow will in general produce singularities, but one may be able to continue the Ricci flow past the singularity by using surgery to change the topology of the manifold.

In 2003 Grigori Perelman sketched a proof of the geometrization conjecture by showing that the Ricci flow can indeed be continued past the singularities, and has the behavior described above.

On 1 July 2010, he turned down the prize, saying that he considers his contribution to proving the Poincaré conjecture to be no greater than that of Richard Hamilton, who introduced the theory of Ricci flow with the aim of attacking the geometrization conjecture.

If the Ricci curvature function Ric ( ξ, ξ ) is constant on the set of unit tangent vectors ξ, the Riemannian manifold is said to have constant Ricci curvature, or to be an Einstein manifold.

Since heat tends to spread through a solid until the body reaches an equilibrium state of constant temperature, Ricci flow may be hoped to produce an equilibrium geometry for a manifold for which the Ricci curvature is constant.

( The Ricci curvature is said to be positive if the Ricci curvature function Ric ( ξ, ξ ) is positive on the set of non-zero tangent vectors ξ.

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL ( n, C ).