Ricci tensor

In differential geometry, the Ricci curvature tensor or simply, Ricci tensorwhich is usually noted by symbols Rab{displaystyle R_{ab}} or Ric, is a bivalent symmetrical tensor obtained as a trace of the curvature tensor, which, like that, can be defined in any variety endowed with an affin connection. It was introduced in 1903 by the Italian mathematician G. Ricci.

If defined in a variety of Riemann, it can be interpreted as a Laplaciano of the metric tensor. Like the metric, the Ricci tensor will be a symmetrical bilineal shape. If both are proportional, Ric=λ λ g{displaystyle {text{Ric}}=lambda g}We'll say the variety is a variety of Einstein.

The Ricci tensor fully determines the curvature tensor, if the corresponding Riemann manifold has dimension n < 4. In general relativity, since spacetime has four dimensions, the Ricci tensor does not completely determine the curvature.

Definition

The Ricci curvature can be expressed in terms of the sectional curvature as follows: for a unit vector v, <R(v), v > is the sum of the sectional curvatures of all the planes traversed by the vector v and a vector of an orthonormal frame containing v (there are n-1 such planes). Here R(v) is the Ricci curvature as a linear operator in the tangent plane, and <.,.> is the metric dot product. The Ricci curvature contains the same information as all such sums over all unit vectors. In dimensions 2 and 3 this is the same as specifying all sectional curvatures or the curvature tensor, but in higher dimensions the Ricci curvature contains less information. For example, Einstein manifolds do not have to have constant curvature in dimensions 4 and more.

Expression in coordinates

Using a natural coordinate system, the Ricci tensor of curvature is equal to:

Rσ σ .. =Rρ ρ σ σ ρ ρ .. =▪ ▪ ρ ρ Interpreter Interpreter .. σ σ ρ ρ − − ▪ ▪ .. Interpreter Interpreter ρ ρ σ σ ρ ρ +Interpreter Interpreter ρ ρ λ λ ρ ρ Interpreter Interpreter .. σ σ λ λ − − Interpreter Interpreter .. λ λ ρ ρ Interpreter Interpreter ρ ρ σ σ λ λ ♪♪

Applications of the Ricci tensor of curvature

Topological invariants

The Ricci curvature can be used to define the Chern classes of a manifold, which are topological invariants (thus independent of the choice of metric). The Ricci curvature is also used in the Ricci flow, where a metric is deformed in the direction of the Ricci curvature. On surfaces, the flow produces a constant Gaussian curvature metric and the uniformization theorem for surfaces follows.

General Relativity

The Ricci curvature plays an important role in general relativity, in fact, the Einstein field equation is written in terms of the Ricci tensor as:

Gμ μ .. =8π π Gc4Tμ μ .. {displaystyle G_{mu nu }={8pi G over c^{4}T_{mu nu }}}}

where: Gμ μ .. {displaystyle G_{mu nu },} It's him. Einstein curvature tensor, Tμ μ .. {displaystyle T_{mu nu } } is the power-moment tensor, c{displaystyle c,} is the speed of light and G{displaystyle G } It's the gravitational constant. Einstein's curvature tensor can be written as:

Gμ μ .. =Rμ μ .. − − 12gμ μ .. R{displaystyle G_{mu nu }=R_{mu nu }-{1 over 2}g_{mu nu }R}

where: Rμ μ .. {displaystyle R_{mu nu } It's Ricci's tensor, gμ μ .. {displaystyle g_{mu nu } is the metric and R{displaystyle R} is the Ricci Curvature Escalator

Global topology and the geometry of positive Ricci curvature

The Myers theorem states that if the Ricci curvature is limited down in a complete variety of Riemann by 0,!}" xmlns="http://www.w3.org/1998/Math/MathML">(n− − 1)k▪0{displaystyle left(n-1right)k 20050,!}0,!}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b91d79d988d048dd80a484badad79e4cf47da576" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:13.453ex; height:2.843ex;"/>, then its diameter is ≤ ≤ π π /k{displaystyle leq pi /{sqrt {k}}}, and variety has to have a finite fundamental group. If the diameter is equal to π π /k{displaystyle pi /{sqrt {k}}}, then the variety is isometric to a constant curvature sphere k.

Bishop-Gromov's inequality states that if the Ricci curvature of a variety m- full dimensions of Riemann is ≥0 then the volume of a ball is smaller or equal to the volume of a ball of the same radius on the m- Euclidian space. Even more, if vp(R){displaystyle v_{p}(R)} denotes the volume of the ball with center p and radio R{displaystyle R} in variety and V(R)=cmRm{displaystyle V(R)=c_{m}R^{m} denotes the volume of the radio ball R in the m- Euclidian space then the function vp(R)/V(R){displaystyle v_{p}(R)/V(R)} It's not growing. (the last inequality can be generalized to an arbitrary curvature and is the dominant point in the Gromov compassionate theorem test.)

The Cheeger-Gromoll partition theorem states that if a complete Riemann manifold with Ricc ≥ 0 has a straight line (ie a minimizing geodesic infinite from both sides) then it is isometric to a R x L, where L is a Riemann manifold.

All the results mentioned above show that the positive Ricci curvature has some geometric meaning, on the contrary, the negative curvature is not so restrictive, in particular as it was shown by Joachim Lohkamp, any manifold admits a negative curvature metric.