FUNÇÃO GRACELI ZETA-GAMA

EQUAÇÃO DE GRACELI.. PARA INTERAÇÕES DE ONDAS E INTERAÇÕES DAS FORÇAS FUNDAMENTAIS.

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

1 / $\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] [-1] =

$\psi ^{*}$ Em mecânica quântica, o OPERADOR DE GRACELI [ $G* =$ $\psi ^{*}$ $TEMA , TODAS AS INTERAÇÕES INCLUINDO TODAS AS INTERAÇÕES DAS FORÇAS FUNDAMENTAIS [AS QUATRO FORÇAS] [ELETROMAGNÉTICA, FORTE, FRACA E GRAVITACIONAL], INTERAÇÕES SPINS-ÓRBITAS, ESTRUTURRA ELETRÔNICA DOS ELEMENTOS QUÍMICOS, TRANSFORMAÇÕES, SISTEMAS DE ONDAS QUÂNTICAS, MOMENTUM MAGNÉTICO de cada elemento químico e partícula, NÍVEIS DE ENERGIA , número quântico , e o sistema GENERALIZADO GRACELI.$

$COMO TAMBÉM ESTÁ RELACIONADO A TODO SISTEMA CATEGORIAL GRACELI, TENSORIAL GRACELI DIMENSIONAL DE GRACELI..$

Em geometria diferencial, o tensor de curvatura de Ricci, ou simplesmente tensor de Ricci, é um tensor bivalente, obtido como um traço do tensor de curvatura. Pode ser pensado como um laplaciano do tensor métrico no caso das variedades de Riemann. Nas dimensões 2 e 3, o tensor de curvatura é determinado totalmente pela curvatura de Ricci. Pode-se pensar na curvatura de Ricci em uma variedade de Riemann como um operador no espaço tangente. Se este operador é simplesmente multiplicado por uma constante, então temos variedade de Einstein. A curvatura de Ricci é proporcional ao tensor métrico neste caso. Esse é mais um caso especial de tensor de Riemann, tendo uma contração em alguns índices seus, como o seguinte exemplo:

{R_{ij}=R_{iaj}^{a}=\partial _{b}\Gamma _{ji}^{b}-\partial _{j}\Gamma _{bi}^{b}+\Gamma _{bc}^{b}\Gamma _{ji}^{c}-\Gamma _{jc}^{b}\Gamma _{bi}^{c}}

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

sendo o símbolo de Christoffel representado por

{\Gamma _{\alpha \beta }^{\rho }={\frac {1}{2}}g^{\rho \lambda }(g_{\alpha \lambda ,\beta }+g_{\lambda \beta ,\alpha }-g_{\beta \alpha ,\lambda })}

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Normalized Ricci flow[edit]

Suppose that $M$ is a compact smooth manifold, and let $g t$ be a Ricci flow for $t$ in the interval $(a, b)$ . Define $Ψ: (a, b) \to (0, \infty)$ so that each of the Riemannian metrics $Ψ(t) g t$ has volume 1; this is possible since $M$ is compact. (More generally, it would be possible if each Riemannian metric $g t$ had finite volume.) Then define $F : (a, b) \to (0, \infty)$ to be the antiderivative of $Ψ$ which vanishes at $a$ . Since $Ψ$ is positive-valued, $F$ is a bijection onto its image $(0, S)$ . Now the Riemannian metrics $G s = Ψ(F -1 (s)) g F -1 (s)$ , defined for parameters $s \in (0, S)$ , satisfy

{\frac {\partial }{\partial s}}G_{s}=-2\operatorname {Ric} ^{G_{s}}+{\frac {2}{n}}{\frac {\int _{M}R^{G_{s}}\,d\mu _{G_{s}}}{\int _{M}d\mu _{G_{s}}}}G_{s}.

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Here

R

denotes scalar curvature. This is called the normalized Ricci flow equation. Thus, with an explicitly defined change of scale

Ψ

and a reparametrization of the parameter values, a Ricci flow can be converted into a normalized Ricci flow. The converse also holds, by reversing the above calculations.

The primary reason for considering the normalized Ricci flow is that it allows a convenient statement of the major convergence theorems for Ricci flow. However, it is not essential to do so, and for virtually all purposes it suffices to consider Ricci flow in its standard form. Moreover, the normalized Ricci flow is not generally meaningful on noncompact manifolds.

Making use of a technique pioneered by Peter Li and Shing-Tung Yau for parabolic differential equations on Riemannian manifolds, Hamilton (1993a) proved the following "Li–Yau inequality."^[5]

Let M be a smooth manifold, and let g_t be a solution of the Ricci flow with t ∈ (0,T) such that each g_t is complete with bounded curvature. Furthermore, suppose that each g_t has nonnegative curvature operator. Then, for any curve γ:[t₁,t₂] → M with [t₁,t₂] ⊂ (0,T), one has ${\frac {d}{dt}}{\big (}R^{g(t)}(\gamma (t)){\big )}+{\frac {R^{g(t)}(\gamma (t))}{t}}+{\frac {1}{2}}\operatorname {Ric} ^{g(t)}(\gamma '(t),\gamma '(t))\geq 0.$

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Perelman (2002) showed the following alternative Li–Yau inequality.

Let M be a smooth closed n-manifold, and let g_t be a solution of the Ricci flow. Consider the backwards heat equation for n-forms, i.e. ∂/∂tω + Δ^g(t)ω = 0; given p ∈ M and t₀ ∈ (0,T), consider the particular solution which, upon integration, converges weakly to the Dirac delta measure as t increases to t₀. Then, for any curve γ:[t₁,t₂] → M with [t₁,t₂] ⊂ (0,T), one has ${\frac {d}{dt}}{\big (}f(\gamma (t),t){\big )}+{\frac {f{\big (}\gamma (t),t{\big )}}{2(t_{0}-t)}}\leq {\frac {R^{g(t)}(\gamma (t))+|\gamma '(t)|_{g(t)}^{2}}{2}}.$
/
$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =
where ω = (4 $π$ (t₀ − t))^−n/2e^−f dμ_g(t).

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

Both of these remarkable inequalities are of profound importance for the proof of the Poincaré conjecture and geometrization conjecture. The terms on the right hand side of Perelman's Li–Yau inequality motivates the definition of his "reduced length" functional, the analysis of which leads to his "noncollapsing theorem." The noncollapsing theorem allows application of Hamilton's compactness theorem (Hamilton 1995) to construct "singularity models," which are Ricci flows on new three-dimensional manifolds. Owing to the Hamilton–Ivey estimate, these new Ricci flows have nonnegative curvature. Hamilton's Li–Yau inequality can then be applied to see that the scalar curvature is, at each point, a nondecreasing (nonnegative) function of time. This is a powerful result that allows many further arguments to go through. In the end, Perelman shows that any of his singularity models is asymptotically like a complete gradient shrinking Ricci soliton, which are completely classified; see the previous section.

See Chow, Lu & Ni (2006, Chapters 10 and 11) for details on Hamilton's Li–Yau inequality; the books Chow et al. (2008) and Müller (2006) contain expositions of both inequalities above.

Em geometria diferencial, o tensor de Einstein (também tensor de traço revertido de Ricci), nomeado em relação a Albert Einstein, é usado para expressar a curvatura de uma variedade de Riemann. Em relatividade geral, o tensor de Einstein aparece nas equações de campo de Einstein para a gravitação descrevendo a curvatura do espaço-tempo.

Definição

O tensor de Einstein $\mathbf {G}$ é um tensor de ordem definido sobre variedades riemannianas. Ele é definido como

\mathbf {G} =\mathbf {R} -{\frac {1}{2}}\mathbf {g} R,

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =

sendo $\mathbf {R}$ o tensor de Ricci, $\mathbf {g}$ o tensor métrico e $R$ o escalar de curvatura de Ricci. Em notação com índices, o tensor de Einstein tem a forma

G_{\mu \nu }=R_{\mu \nu }-{1 \over 2}g_{\mu \nu }R.

$\psi ^{*}$ [ $\psi ^{*}$ ${\boldsymbol {\mu }}$ ] $\lambda$ ω ${\mathcal {T}}$ , $\sigma$ , $\psi$ $-i\hbar {\frac {\partial }{\partial x_{n}}}$ [x,t] ] =