Ricci solitons

These days I am looking at all this area of mathematical research born with Richard Hamilton and put at maturity with the works of Grisha Perelman. As all of you surely know the conclusion was that the Thurston conjecture, implying Poincare’ conjecture, is a theorem. These results present the shocking aspect of a deep truth waiting for an understanding by physicists and, I think that this comes out unexpectedly, statisticians (do you know Fischer information matrix and Cramer-Rao bound?).

One of the most shocking concept mathematicians introduced working with Ricci flow is a Ricci soliton. I will use some mathematics to explain this. A Ricci flow is given by

\frac{\partial g_{ik}}{\partial t}= -2R_{ik}

a Ricci soliton is a metric solving the equation

R_{ik}-L_X g_{ik} = \Lambda g_{ik}

where I have used an awkward notation for the Lie derivative along a field X but if this field is a scalar than one has a gradient soliton. I think that all of you will recognize these equations that for a Lorentzian metric are just Einstein equations in vacuum with a cosmological constant! Now, I have found a beautiful paper about all this question on arxiv (see here). This paper gives the first meaningful application to physics of this striking concept. Ricci solitons are resembling a kind of behavior of the metric under the flow that can be expanding, collapsing or static depending on the cosmological constant.

As time goes by we learn something deeper about Einstein equations. Their very nature seems rooted in quite recent concepts coming from differential geometry and it is my personal view that whatever quantum gravity theory we will formulate, these are the questions we have to cope with.

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6 Responses to Ricci solitons

  1. Daniel de França MTd2 says:

    Hi,

    More articles from the same group that I mentinoned here:

    https://marcofrasca.wordpress.com/2009/01/15/ted-jacobson-and-quantum-gravity/#comment-462

    http://arxiv.org/abs/0808.3126v2

    Ricci flow, quantum mechanics and gravity
    Authors: J.M. Isidro, J.L.G. Santander, P. Fernandez de Cordoba
    (Submitted on 18 Aug 2008)

    Abstract: It has been argued that, underlying any given quantum-mechanical model, there exists at least one deterministic system that reproduces, after prequantisation, the given quantum dynamics. For a quantum mechanics with a complex d-dimensional Hilbert space, the Lie group SU(d) represents classical canonical transformations on the projective space CP^{d-1} of quantum states. Let R stand for the Ricci flow of the manifold SU(d-1) down to one point, and let P denote the projection from the Hopf bundle onto its base CP^{d-1}. Then the underlying deterministic model we propose here is the Lie group SU(d), acted on by the operation PR. Finally we comment on some possible consequences that our model may have on a quantum theory of gravity.

    http://arxiv.org/abs/0808.2351v1

    A note on the quantum-mechanical Ricci flow
    Authors: J.M. Isidro, J.L.G. Santander, P. Fernandez de Cordoba
    (Submitted on 20 Aug 2008 (v1), last revised 8 Dec 2008 (this version, v2))

    Abstract: We obtain Schroedinger quantum mechanics from Perelman’s functional and from the Ricci flow equations of a conformally flat Riemannian metric on a closed 2-dimensional configuration space. We explore links with the recently discussed “emergent quantum mechanics”.

    http://arxiv.org/abs/0810.2236v3

    The Ricci flow on Riemann surfaces
    Authors: S. Abraham, P. Fernandez de Cordoba, J.M. Isidro, J.L.G. Santander
    (Submitted on 13 Oct 2008 (v1), last revised 8 Dec 2008 (this version, v3))

    Abstract: We establish a 1-to-1 relation between metrics on compact Riemann surfaces without boundary, and mechanical systems having those surfaces as configuration spaces.

  2. Kea says:

    Hear, hear! I look forward to more posts as you study this further.

  3. mfrasca says:

    Hi Daniel,

    Thanks a lot for the very interesting papers you pointed out. I have taken a look at them and I should say that we are just starting to uncover a lot of interesting new views in physics starting from this new chapter of mathematics. We have just scratched the surface.

    Hi Kea,

    I hope to deepen this matter a lot more in the future.

    Marco

  4. Daniel de França MTd2 says:

    Marco,

    did you know that the most complicated manifolds are not those of n4, but exactly n=4? It sounds weird, but that happens for a variety of reasons. I will point out a certain book.

    World Scientific – Asselmeyer-Maluga T., Brans C.H. – Exotic Smoothness and Physics; Differential Topology and Spacetime Models (2007)

    You can easily find it on emule. Also, look for articles from Asselmeyer and Brans at arxiv.org.

    There are the books “The Wild World of 4 Manifolds” from Scorpan and “Kirby Calculus”, from, well, forgot for now….

    I will tell you more later.

    Daniel.

  5. mfrasca says:

    Daniel,

    Let me say that I appreciate a lot this kind of informative comments. I will look at the book you cite. It is my personal view that there is a lot more to be said about quantum gravity than our current theories contain.

    Marco

  6. Musavvir Ali says:

    Marco its nice to see a person working in same field Ricci soliton. I have thouroughly read the paper you hint by MM akbar and E woolger. I have also seen a nice correspondance between Ricci solitons and symmetries of spacetime manifold. I am searching some concrete examples, if I got then send you. You can also help me for the same.

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