Symmetries of Ricci flows
-
Enrique López
,
Stylianos Dimas
Abstract
In the present work, we find the Lie point symmetries of the Ricci flow on an n-dimensional manifold, and we introduce a method in order to reutilize these symmetries to obtain the Lie point symmetries of particular metrics. We apply this method to retrieve the Lie point symmetries of the Einstein equations (seen as a “static” Ricci flow) and of some particular types of metrics of interest, such as, on warped products of manifolds. Finally, we use the symmetries found to obtain invariant solutions of the Ricci flow for the particular families of metrics considered.
1 Introduction
Hamilton, in 1982 [10], introduced the Ricci flow: a nonlinear geometric evolution equation in which one starting with a smooth Riemannian manifold
where
It is well known that in contrast to the theory of linear differential equations – although the great progress in the past seven decades – there is still no unitary theory for nonlinear differential equations and systems. Nevertheless, various effective approaches have been developed: compact operators and operators of monotone type; topological degree theories; fixed point theory and modern variational principles; critical point theory; Morse theory and critical groups, treated from a general qualitative point of view [15,24]; Rellich and Pohozaev identities via a Noetherian approach [5–7]; analytical methods for constructing exact solutions of differential equations [18], etc.
We believe that the symmetry approach based on the application of Lie symmetry methods for differential equations may be fruitful in the nonlinear analysis. We note that Sophus Lie originally introduced the general theory of Lie transformation groups in the last part of the nineteenth century, specifically to study differential equations. After a period of dormancy, Lie’s ideas resurfaced, beginning with the pioneering works of L. Ovsyannikov and his collaborators. It is no exaggeration to state that the great array of applications of Lie group methods to nonlinear differential equations has been one of the great successes of the last 60 years. There is currently a vast literature in this area, and many of the basic methods and applications can be found in the studies of Olver [20,21] and Ovsiannikov [22] and the three-volume handbook by Ibragimov [13]. Some of the directions being developed include the construction of explicit symmetric solutions through symmetry reduction, the connection between symmetries and conservation laws via the Noether’s theorem for problems admitting variational structures, construction, and classification of differential invariants, invariant differential equations and variational principles, separation of variables and integrability of both linear and nonlinear partial differential equations, asymptotic behavior of solutions for long times and near blow-up using symmetric solutions, and the design of symmetry-preserving numerical algorithms.
We recall that the symmetries of a differential equation transform solutions of the equation to other solutions. One of the main benefits of this theory is that by following a completely algorithmic procedure, one is able to determine the symmetries of a differential equation or systems of differential equations. They comprise a structural property of the equation – in the essence they are equation’s DNA. The knowledge of the symmetries of an equation enables one to use them for a variety of purposes, from obtaining analytical solutions and reducing its order to finding of integrating factors and conservation laws as mentioned earlier. In fact, many, if not all, of the different empirical methods for solving ordinary differential equations (ODE) we have learned from standard courses at the undergraduate level emerge from symmetry. For instance, given a Lie point symmetry of a first-order ODE, we can immediately and explicitly obtain an integrating factor by a formula obtained by S. Lie. In this regard, we remind the words of Nail Ibragimov that
One of the most remarkable achievements of Lie was the discovery that the majority of known methods of integration of ordinary differential equations, which seemed up to that time artificial and internally disconnected, could be derived in a unified manner using the theory of groups. Moreover, Lie provided a classification of all ordinary differential equations of arbitrary order in terms of the symmetry groups they admit and thus described the whole collection of equations for which integration or lowering of the order could be effected by group-theoretical methods. However, these and other very valuable results he obtained could not for a long time be widely disseminated and remained known to only a few. It could be said that this is the state of affairs today with methods of solution of problems of mathematical physics: many of these are of a group-theoretical nature, but are presented as a result of a lucky guess [12].
All in all, symmetries play a far-reaching role in the analysis of differential equations: they allow us to recover the full picture from partial information. For this reason, it is not an uncommon practice to employ symmetries for constructing new solutions from known ones or for reducing a differential equation. At this point, it is worth mentioning that for the main bulk of calculations, we used the symbolic package SYM [9].
So far, we only have a description of the Lie point symmetries of the Ricci flow for the two-dimensional case, see Cimpoiasu and Constantinescu [8] and Wang [29]. The goal of this article is to carry out a complete group classification of the Ricci flow on
Theorem 1
The Lie algebra of the classical symmetries of the Ricci flow on a Riemannian manifold
where
This article is organized as follows: in Section 2, we give a brief introduction of all the notions utilized in the rest of the article and illustrate the key method we employ for obtaining the Lie point symmetries of the Ricci flow for particular families of metrics from the Lie algebra of Theorem 1. Our main result resides in Section 3, where we determine the Lie point symmetries of the Ricci flow for the
2 Preliminaries
In this section, we will briefly introduce the general concept of Lie point symmetry of differential equations. For a detailed approach, we recommend reading [11,13,20,22]. We restrict our attention to connected local Lie groups of symmetries, leaving aside discrete symmetries.
Let assume a system of differential equations of
involving the derivatives of
If
determines a sub-manifold of the jet space. Henceforth, we will consider zero to be a regular value.
A smooth solution of the given system of differential equations is a smooth vector function
This is just a restatement of the fact that the derivatives
A Lie point symmetry group of this system is a local group of transformations, which maps solutions to solutions. That is, if
The tangent space of a local group of transformations is a Lie algebra – its vectors are often called infinitesimal generators. With starting point a Lie algebra by employing its exponential mapping, we can retrieve the local group of transformations with each element of the Lie algebra giving rise to a local subgroup of transformations called (local) one-parameter group. Through this local isomorphism, it is common practice to identify a Lie point symmetry with an infinitesimal generator.
Theorem 2
[20, Theorem 2.31]. Suppose that
is a system of differential equations over
for every infinitesimal generator
From these infinitesimal generators, it is possible to reduce the number of independent variables of the given system of equations and even obtain solutions. The solutions thus obtained are called invariant solutions of the infinitesimal generator employed. Thus, the natural question that arises once the set of symmetries has been calculated is how to obtain all the invariant solutions. This leads to the concept of optimal system, see [20, Proposition 3.6].
The main objective of our study is the Lie point symmetries of the Ricci flow for a general metric. A difficult and laborious task indeed! However, from these symmetries – by a method we developed – we will be able to retrieve the symmetries for any particular family of metrics by restricting the generic Lie algebra to the metric at hand, thereby eliminating the need to repeat each time the same copious process for finding the Lie point symmetries. This method is based on a particular form of the infinitesimal generator known as the canonical form, or simply, the characteristic.
Recall that given an infinitesimal generator
its canonical form is the vector
where
In the following example, we illustrate the method.
Example 1
We wish to obtain the Ricci flow symmetries of the metric
from the symmetries of Theorem 1. Let
where
where
Now, looking at the metric, it is easy to see that we need to impose the following restrictions to the generic metric of dimension 2:
We are ready to give the restrictions that we need to impose to the Lie algebra of Theorem 1 for
Therefore, the Lie algebra for the restricted problem is:
where
3 The Lie point symmetries of the Ricci flow
In this section, we compute the Lie point symmetries of the Ricci flow for the
By direct calculation, using SYM, of the Lie point symmetries for the cases
Conjecture 1
The Lie algebra of the classical symmetries, i.e., Lie point symmetries, of the Ricci flow on a Riemannian manifold
where
It is easy to verify that the Lie algebra spanned by (1) leaves invariant the Ricci flow for
The symmetry
As for
we obtain that
and
Finally, let us turn to the infinite ideal
Let
(2)We wish to analyze
So, equation (2) becomes
Therefore, as we have claimed, the symmetries
Proof
Let us suppose that
is the infinitesimal generator of a Lie point symmetry of the Ricci flow. Observe that the second summation is restricted so that the mixed derivatives of g are not counted twice.
Using this generic form, we can arrive at the determining equations. At this point, we employ the idea of Marchildon [17] to solve them. First, we write the Ricci flow as
where
and
At this point it is advantageous – for the sake of clarity – to define the tensor
Since
and
In particular,
We need to express the partial derivatives of the Ricci tensor with respect to
see [17]. Note that for any symmetric tensor
The second prolongation of
Therefore, to obtain the linearization conditions, we first need to expand the equations
where
The functions
see Olver [20, Theorem 2.36], where
Hence,
and
Finally, we obtain the linearized symmetry conditions – having in mind that the metric must also satisfy the Ricci flow equation,
The following is the strategy that we will abide by: from the linearized symmetry conditions, we will extract the determining equations starting with the terms involving derivatives of the metric tensor that do not appear in the Ricci flow equation, then continue with terms that involve higher-order derivatives of the metric, and finally those involving the derivatives of the low-order metric tensor since the corresponding determining equations are usually much easier to solve in this order.
Using the expressions of
where
Renaming the indexes on the right-hand side of equation (9) as
We are now ready to extract some of the coefficients equations (3)–(5).
From the
Rearranging the previous equation until we have the same index in the derivatives of
Case
Thus,
for all
Observe that, the terms
Since, in the Ricci flow, there are no terms of the form
Rearranging equation (12), we obtain
Case
i.e.,
for all
From the
Rearranging indices, one can show that the previous equations become
Case
Thus,
for all
Now we group the terms that contain only second spatial derivatives of the metric tensor. However, since we want that
Rearranging the indices, we obtain
Case
and in consequence
with
Case
Using equation (16), we obtain
with
Case
Using equation (17), we obtain from equation (18) that
for
Note that the only way for
Case
Furthermore, if
Finally, if
From equations (19) and (21), we obtain
with
Case
Therefore,
with
Case
By equations (23) and (24), it follows that
with
Case
and by differentiating the above equation with respect to
Note that the expressions in parentheses do not depend on
On the other hand, since
for all
with
Case
Putting (27) and (25), we obtain
We can write
Case
Using equations (26) and (28), we obtain
Differentiating with respect to
Combining equations (29) and (30), we can conclude that
Case
Using equations (26) and (28), this above equation becomes
By equation (31), we have
for all
Differentiating with respect to
for all fixed
We can conclude that
for all
Now, we differentiate equation (32) with respect to
for all fixed
Therefore,
for all
for all
From the
On the other hand, note that
see equations (26) and (28). Using equations (26) and (28), equation (35) becomes
Rearranging indices and considering the symmetry
Case
for
for all
for all
Since
Case
for all
for all
Terms that do not contain derivatives of the metric tensor: we have the following determining equation:
Hence,
Thus,
Therefore, (38) becomes
where
From the
for all fixed
Case
thus
By differentiating it for
From this equation, we have
Gathering everything together, we can conclude that a sufficient condition for
Hence, we have proved Theorem 1.□
Now, we classify the finite-dimensional algebra Theorem 1
Proposition 1
The one-dimensional optimal system associated with the Lie algebra
Proof
See example 10.1 in [11].□
Observation 1
The Ricci flow viewed as a mapping of a suitable jet space is analytic. Thus, we have the guarantee that the symmetries obtained through the infinitesimal criterion are the maximum continuous symmetry group, see Corollary 2.74 in [20].
4 The Ricci flow for particular families of metrics
4.1 The Einstein equations
In this section, we obtain the Lie point symmetries of the vacuum Einstein equations retrieving them from the ones of the Ricci flow.
Proposition 2
The Lie algebra of the Lie point symmetries of the vacuum Einstein equations in
where
Proof
Let
where
and employ the method illustrated in Example 1. That being so, we need to consider the following restriction:
or equivalently,
from which we arrive at the desired proposition.□
Observation 2
These symmetries in
In the following sections, we apply this method to find the symmetries of the Ricci flow for particular metric families.
4.2 Ricci flow on warped product manifolds
Given two Riemannian manifolds
We will study the Ricci flow on the warped product
and by abuse of notation, we write
where
We know that the Ricci flow preserves the isometry group of the initial Riemannian manifold. For warped metrics, this means that the Ricci flow preserves the warped structure (see [19]). As a consequence, we can write
Consider now the smooth fields
and by the Ricci flow, we have
Thus,
Finally, we have
with
where
Theorem 3
The Lie algebra of the Lie point symmetries of the Ricci flow of equation (40), with
where
with
Proof
We begin by taking a symmetry of the Ricci flow
where
Having in mind (40), we see that we need to apply the method on the restrictions
under the restrictions described above, we obtain the conditions
for all
In addition, we have that
for
with
so
for all
For
In this case,
for all
in this last equation
Thus,
Putting all of the above together
with
Now we turn our attention to the special case where the fiber is the Euclidean space endowed with the canonical metric.
Theorem 4
The Lie algebra of the Lie point symmetries of the Ricci Flow of (40), with
where
with
Proof
Following the previous proof, we have that
for
where
with
Corollary 1 is an immediate consequence of Theorem 4.
Corollary 1
The Lie algebra of the Lie point symmetries of the Ricci flow on
is spanned by the following set of vectors:
where
with
Proof
The result follows immediately through the first part of the proof of Theorem 3.□
4.3 Ricci flow on doubly warped product manifolds
We now restrict our attention to flows evolving from a fixed initial metric that is cohomogeneous with respect to the natural
where
Theorem 5
The Lie algebra of the Lie point symmetries of the Ricci flow on
is spanned by the following infinitesimal generators:
with
Proof
Similar to the proof of Theorem 3, we start by taking a symmetry from Theorem 1, i.e.,
where
Using the metric form (45), we see that we need to apply the method on the restrictions
and under the restrictions described above and proceeding analogously to the proof of Theorem 3, we obtain
From this, we can conclude the statement of the theorem.□
5 Invariant solutions
5.1 Invariant solutions for warped product manifolds
First, we need to explicitly express the Ricci flow equations for the metric tensor (40). We know that
for all
From conformal theory, we know that
and
see Theorem 1.159 in [3]. Thus,
On the other hand,
Again, from conformal theory, we have
Therefore,
That being done, the family of metrics (40) satisfies the Ricci flow equation if and only if the following system is satisfied:
for all
Then, we use the symmetries found in Section 4.2 to reduce the system of equation (46).
Note that
where we have chosen
Solving the system (47) by the method of characteristics, we obtain
where
for all
where
Integrating it, we obtain
Observe that this solution satisfies identically the first equation of the system (49). Furthermore, the solution suggests the following parameterization:
Therefore,
So, having all the solutions of system (49) at hand, when
when
5.2 Invariant solutions for doubly warped product manifolds
First, we need to express explicitly the Ricci flow equations. In this case, see [26],
By the same argument as in Section 5.1, we will make use of the following symmetry:
where
Solving system (51), we obtain
Inserting these expressions of
Taking inspiration from the procedure to obtain the solutions for the case of single warped product manifolds, we start by parametrizing
The system (52) becomes
Once more, by studying the form of the solutions found in Section 5.1, we start looking for particular solutions of the same form. This hypothesis was fruitful, the following functions
for
for
Hence, our study yields the following special solutions:
where
6 Conclusion and discussion
In the present work, we determine the Lie point symmetries of the Ricci flow in arbitrary dimensions. By using their algebraic properties, we are able to “recycle” them in order to expeditiously obtain the Lie point symmetries of the Ricci flow for a particular family of metrics. A task that can be very challenging if one starts each time from scratch. Indeed, even in the reasonable case of studying a four-dimensional metric to obtain its Lie point symmetries one had to solve a system of determining equations that involves about two million PDE!
Unfortunately, classical symmetries, being a very broad notion that can be applied virtually to any kind of system of differential equations, have a very serious drawback; they usually will not help us to obtain the kinds of solutions that we are looking for. This is apparent in our case: the infinite-dimensional sub algebra merely points to the tensorial nature of our system, while the finite one says, on the one hand, that the system is autonomous and, on the other hand, ushers us to consider separable solutions. Solutions that are known to hold if and only if the initial manifold is an Einstein one. A fact that limits our choices regarding the initial condition to be considered.
One possible way out is by considering more “exotic” kinds of symmetries, like the nonclassical ones, see Section 9.3 in [11]. In a word, nonclassical symmetries are symmetries that are admitted only by specific families of solutions of a differential equation and not by the differential equation itself, as is the case with the classical symmetries. Although our initial studies were promising, none of the nonclassical symmetries found yielded nontrivial solutions. This do not come as a surprise since the systems of differential equations involved are now nonlinear. Therefore, a more exhausting classification must be carried over to obtain nonclassical symmetries that can yield solutions of the Ricci flow problem of some interest.
Acknowledgements
Enrique López thanks the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) - Brazil, for the financial support (Finance Code 001).
-
Conflict of interest: The authors declare that they have no conflict of interest.
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- Bifurcation diagrams of one-dimensional Kirchhoff-type equations
- Standing wave solution for the generalized Jackiw-Pi model
- Weak-strong uniqueness and energy-variational solutions for a class of viscoelastoplastic fluid models
- Eigenvalue inequalities for the buckling problem of the drifting Laplacian of arbitrary order
- Homoclinic solutions for a differential inclusion system involving the p(t)-Laplacian
- Equivalence between a time-fractional and an integer-order gradient flow: The memory effect reflected in the energy
- Bautin bifurcation with additive noise
- Small solitons and multisolitons in the generalized Davey-Stewartson system
- Nonstationary Poiseuille flow of a non-Newtonian fluid with the shear rate-dependent viscosity
- A global compactness result with applications to a Hardy-Sobolev critical elliptic system involving coupled perturbation terms
- On a strongly damped semilinear wave equation with time-varying source and singular dissipation
- Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals
- Stability of stationary solutions to the three-dimensional Navier-Stokes equations with surface tension
- Uniform decay estimates for the semi-linear wave equation with locally distributed mixed-type damping via arbitrary local viscoelastic versus frictional dissipative effects
- Symmetry and nonsymmetry of minimal action sign-changing solutions for the Choquard system
- Periodic and quasi-periodic solutions of a four-dimensional singular differential system describing the motion of vortices
- Multiple nontrivial solutions of superlinear fractional Laplace equations without (AR) condition
- Existence and blow-up of solutions in Hénon-type heat equation with exponential nonlinearity
- Existence and concentration behavior of positive solutions to Schrödinger-Poisson-Slater equations
- On the fractional Korn inequality in bounded domains: Counterexamples to the case ps < 1
- Gradient estimates for nonlinear elliptic equations involving the Witten Laplacian on smooth metric measure spaces and implications
- Dynamical analysis of a reaction–diffusion mosquito-borne model in a spatially heterogeneous environment
- Existence and multiplicity of solutions for a quasilinear system with locally superlinear condition
- Nontrivial solution for Klein-Gordon equation coupled with Born-Infeld theory with critical growth
- Modeling Wolbachia infection frequency in mosquito populations via a continuous periodic switching model
- Infinitely many localized semiclassical states for nonlinear Kirchhoff-type equation
- Fujita-type theorems for a quasilinear parabolic differential inequality with weighted nonlocal source term
- Approximations of center manifolds for delay stochastic differential equations with additive noise
- Periodic solutions to a class of distributed delay differential equations via variational methods
- Groundstate for the Schrödinger-Poisson-Slater equation involving the Coulomb-Sobolev critical exponent
- Existence of nontrivial solutions for the Klein-Gordon-Maxwell system with Berestycki-Lions conditions
- Global Sobolev regular solution for Boussinesq system
- Normalized solutions for the p-Laplacian equation with a trapping potential
- Nonlinear elliptic–parabolic problem involving p-Dirichlet-to-Neumann operator with critical exponent
- Blow-up for compressible Euler system with space-dependent damping in 1-D
- High energy solutions of general Kirchhoff type equations without the Ambrosetti-Rabinowitz type condition
- On the dynamics of grounded shallow ice sheets: Modeling and analysis
- A survey on some vanishing viscosity limit results
- Blow-up for logarithmic viscoelastic equations with delay and acoustic boundary conditions
- Generalized Liouville theorem for viscosity solutions to a singular Monge-Ampère equation
- Front propagation in a double degenerate equation with delay
- Positive solutions for a class of singular (p, q)-equations
- Higher integrability for anisotropic parabolic systems of p-Laplace type
- The Poincaré map of degenerate monodromic singularities with Puiseux inverse integrating factor
- On a system of multi-component Ginzburg-Landau vortices
- Existence and concentration of solutions to Kirchhoff-type equations in ℝ2 with steep potential well vanishing at infinity and exponential critical nonlinearities
- Multiplicity results for fractional Schrödinger-Kirchhoff systems involving critical nonlinearities
- On double phase Kirchhoff problems with singular nonlinearity
- Estimates for eigenvalues of the Neumann and Steklov problems
- Nodal solutions with a prescribed number of nodes for the Kirchhoff-type problem with an asymptotically cubic term
- Dirichlet problems involving the Hardy-Leray operators with multiple polars
- Incompressible limit for compressible viscoelastic flows with large velocity
- Characterizations for the genuine Calderón-Zygmund operators and commutators on generalized Orlicz-Morrey spaces
- Non-local gradients in bounded domains motivated by continuum mechanics: Fundamental theorem of calculus and embeddings
- Noncoercive parabolic obstacle problems
- Touchdown solutions in general MEMS models
- Existence and nonexistence of nontrivial solutions for the Schrödinger-Poisson system with zero mass potential
- Short-time existence of a quasi-stationary fluid–structure interaction problem for plaque growth
- Fine bounds for best constants of fractional subcritical Sobolev embeddings and applications to nonlocal PDEs
- Symmetries of Ricci flows
- Asymptotic behavior of solutions of a second-order nonlinear discrete equation of Emden-Fowler type
- On the topological gradient method for an inverse problem resolution
- Supersolutions to nonautonomous Choquard equations in general domains
- Uniform complex time heat Kernel estimates without Gaussian bounds
- Global existence for time-dependent damped wave equations with nonlinear memory
- Normalized solutions for a critical fractional Choquard equation with a nonlocal perturbation
- Reversed Hardy-Littlewood-Sobolev inequalities with weights on the Heisenberg group
- Lamé system with weak damping and nonlinear time-varying delay
- Global well posedness for the semilinear edge-degenerate parabolic equations on singular manifolds
- Blowup in L1(Ω)-norm and global existence for time-fractional diffusion equations with polynomial semilinear terms
- Boundary regularity results for minimisers of convex functionals with (p, q)-growth
- Parametric singular double phase Dirichlet problems
- Special Issue on Nonlinear analysis: Perspectives and synergies
- Editorial to Special issue “Nonlinear analysis: Perspectives and synergies”
- Identification of discontinuous parameters in double phase obstacle problems
- Global boundedness to a 3D chemotaxis-Stokes system with porous medium cell diffusion and general sensitivity
- On regular solutions to compressible radiation hydrodynamic equations with far field vacuum
- On Cauchy problem for fractional parabolic-elliptic Keller-Segel model
- The evolution of immersed locally convex plane curves driven by anisotropic curvature flow
- Stability of combination of rarefaction waves with viscous contact wave for compressible Navier-Stokes equations with temperature-dependent transport coefficients and large data
- On concave perturbations of a periodic elliptic problem in R2 involving critical exponential growth
- Existence of solutions for a double-phase variable exponent equation without the Ambrosetti-Rabinowitz condition
