Einstein Finsler Metrics and Ricci flow Nasrin Sadeghzadeh - PowerPoint PPT Presentation

Einstein Finsler Metrics and Ricci flow Nasrin Sadeghzadeh University of Qom, Iran Jun 2012 XXXI Workshop on Geometric Methods in Physics

Outline - A Survey of Einstein metrics, - A brief explanation of Ricci flow and its extension to Finsler Geometry, Ricci flow is used to - Study the Existence of Einstein Finsler metric of non-constant Ricci scalar. XXXI Workshop on Geometric Methods in Physics

Einstein Metrics ( Riemann and Finsler ) XXXI Workshop on Geometric Methods in Physics

History 1  2  Ric Sg T XXXI Workshop on Geometric Methods in Physics

History 1  Ric Sg 2 XXXI Workshop on Geometric Methods in Physics

History XXXI Workshop on Geometric Methods in Physics

Definition of Einstein Riemannian Manifolds XXXI Workshop on Geometric Methods in Physics

Other Interpretation XXXI Workshop on Geometric Methods in Physics

Relation with other properties (Rie. Schur Lemma) For manifolds of dimension up to three, Einstein Riemannian metrics are precisely the same as constant (sectional) curvature metrics XXXI Workshop on Geometric Methods in Physics

Einstein Finsler metric XXXI Workshop on Geometric Methods in Physics

Einstein Finsler metrics Akbar-Zadeh in his paper in titled [1995] “Generalized Einstein Manifolds” states Einstein Finsler manifolds are critical points of scalar functional, the same as Riemannian case. However the integrand function is not the same Riemannian case. XXXI Workshop on Geometric Methods in Physics

Finsler Ricci Tensor 1 1   p q H { H y y ( , )} { H y y } , ij i j pq i j . y . y . y . y 2 2 H  r H i rj ij y u  F XXXI Workshop on Geometric Methods in Physics

Other Interpretation XXXI Workshop on Geometric Methods in Physics

Question? XXXI Workshop on Geometric Methods in Physics

To Answer We use Ricci flow as a tool to investigate the answer of the question . XXXI Workshop on Geometric Methods in Physics

What is Ricci flow? XXXI Workshop on Geometric Methods in Physics

Basic Question (Riemannian case) How can we distinguish the three-dimensional sphere from the other three-dimensional manifolds? XXXI Workshop on Geometric Methods in Physics

History (Riemannian case) At the beginning of the 20th century, Henri Poincaré was working on the foundations of topology , announced his conjecture Every simply connected compact 3 - manifold ) without boundary) is homeomorphic to a 3 - sphere . Poincar é 's conjecture became the base of Ricci flow equation. XXXI Workshop on Geometric Methods in Physics

History (Riemannian case) Hamilton's program and Perelman's solution Hamilton's program was started in his paper in 1982, which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture.

History (Riemannian case) The actual solution wasn't found until Grigori Perelman (of the Steklov institute of Mathematics, Saint petersburg) published his papers using many ideas from Hamilton's work (Ricci flow equation with surjery) . On August 22 , 2006 , the ICM awarded Perelman the Fields Medal for his work on the conjecture, but Perelman refused the medal .

Perelman’s Proof He put a Riemannian metric on the unknown simply connected closed 3 - manifold . The idea is to try to improve this metric. The metric is improved using the Ricci flow equations ;  g   ij 2 Ric ,  ij t where g is the metric and R its Ricci curvature, and one hopes that as the time t increases, the manifold becomes easier to understand .

Perelman ’ s theorem Every closed 3-manifold which admit a metric of positive Ricci curvature also admit a metric of constant positive sectional curvature. XXXI Workshop on Geometric Methods in Physics

Ricci flow & heat equation Somewhat like the heat equation  f   2 f  t except nonlinear.  g   ij Ric  ij t Heat equation evolves a function & Ricci flow evolves a Riemannian metric. XXXI Workshop on Geometric Methods in Physics

Why Normal Ricci flow equation?? Hamilton found that, sometimes the scalar curvature explodes to +∞ at each point at the same time T and with the same speed. Then He showed that it is necessary to form a normalization that makes the volume constant. XXXI Workshop on Geometric Methods in Physics

What is Normal Ricci flow equation    Vol g ( ) dV 1 ij M  g 2      ij 2 Ric dV .  ij t n M XXXI Workshop on Geometric Methods in Physics

Unnormalized Ricci flow Normalized Ricci flow XXXI Workshop on Geometric Methods in Physics

Ricci flow in Finsler geometry XXXI Workshop on Geometric Methods in Physics

Chern question Does every manifold admit an Einstein Finsler metric or a Finsler metric of constant flag curvature? It is hoped that the Ricci flow in Finsler geometry eventually proves to be viable for addressing Chern's question. XXXI Workshop on Geometric Methods in Physics

Why is there Ricci flow equation in Finsler space? In principle, the same equation can be used in the Finsler setting, g Because both and have been Ric ij ij generalized to that broader framework, albeit gaining a dependence in the process. y XXXI Workshop on Geometric Methods in Physics

Un-normalized equation Bao [2007] have stated a scalar equation instead of this tensor evolution equation. i j l l He contracted the equation with , and via Euler’s theorem is gotten      log F Ric , F t ( 0) F t 0 XXXI Workshop on Geometric Methods in Physics

Normalized equation XXXI Workshop on Geometric Methods in Physics XXXI Workshop on Geometric Methods in Physics

Normalized Equation     log F Ric C t ( ), t 1    C t ( ) RicdV Avg Ric ( ). SM Vol SM SM XXXI Workshop on Geometric Methods in Physics XXXI Workshop on Geometric Methods in Physics

Tensor Ricci flow equation Finsler Ricci flow equation in the tensor form is the same as Riemannian case. It can be used the Akbar- Zadeh’s version of Ricci tensors as  m k l Ric ( R y y ) . k i j ij ml . y . y XXXI Workshop on Geometric Methods in Physics

Einstein Metric of Non-Constant Ricci Scalar XXXI Workshop on Geometric Methods in Physics

Fixed Point of Ricci Flow Equation XXXI Workshop on Geometric Methods in Physics

Finsler self-similar Solution XXXI Workshop on Geometric Methods in Physics

Finsler Ricci Solitons XXXI Workshop on Geometric Methods in Physics

Equivalency of these two definitions XXXI Workshop on Geometric Methods in Physics

Equivalency of these two definition XXXI Workshop on Geometric Methods in Physics

Einstein Finsler metrics of non-constant Ricci Scalar XXXI Workshop on Geometric Methods in Physics

Let - 𝐺 0 be a projrctively flat Finsler metric on 𝑁 , - 𝐺 𝑢 = ℎ(𝑢, 𝑦)𝐺 0 , where ℎ ≔ ℎ 𝑢, 𝑦 is a positive continues function on 𝑁. Then 𝒊 ′ 𝑮 𝒖 is Ricci constant iff ( 𝒊 ) ;𝒚 𝒎 = 𝟏. XXXI Workshop on Geometric Methods in Physics

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Final Equation (PDE) XXXI Workshop on Geometric Methods in Physics

Some References • Akbar-Zadeh H. , Generalized Einstein manifolds, J. Geom. Phys.17(1995), 342-380. • Bao D. , On two curvature-driven problems in Riemann-Finsler geometry, Advanced Studies in Pure Mathematics XX, 2007. • Cao H. -D. and Zhu X. -P. , Hamilton –Perelman’s proof of the Poincar’e conjecture and the geometrization conjecture, Asian J. Math. 10 (2006), 165 – 495; arXiv: math. DG/ 0612069. XXXI Workshop on Geometric Methods in Physics

Some References • Hamilton R. S. , Four Manifolds with Positive Curvature Operator, J. Diff. Geom. 24, 153-179, 1986. • SadeghZadeh N. and Razavi A. , Ricci Flow equation on C-reducible metrics, International Journal of Geometric Methods in Modern Physics, (2011), DOI No: 10.1142/S0219887811005385. • VACARU S. , ON GENERAL SOLUTIONS OF EINSTEIN EQUATIONS, International Journal of Geometric Methods in Modern PhysicsVol. 8, No. 1 (2011) 9 – 21c. XXXI Workshop on Geometric Methods in Physics

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