Manifold Micro-Surgery with Linearly Nearly Euclidean Metrics

1 INTRODUCTION . In general relativity ( Wald , 2010 ) , a complete Riemannian manifold ( M , g ) endowed with a linearly nearly flat spacetime metric gij is considered for linearized gravity to solve the Newtonian limit . The form of this metric is gij = ηij + γij , where ηij represents a flat Minkowski metric whose background is special relativity and γij is small from ηij . An adequate definition of “ smallness ” in this context is that the components of γij are much smaller than 1 in some global inertial coordinate system of ηij . Now , let us step out of the physical world and give a similar metric gij = δij + γij in Riemannian n-manifold ( Mn , g ) , i.e . the linearly nearly Euclidean metric , where δij represents a flat Euclidean metric and γij is small from δij . A natural problem for such a linearly nearly Euclidean metric is : how does the metric evolve over time with respect to the Ricci flow while ensuring the constant topological structure ? Let us review some stability analyses of different manifolds along with the Ricci flow . For the Riemannian n-dimensional manifold ( Mn , g ) that is isometric to the Euclidean ndimensional space ( Rn , δ ) , Schnürer et al . ( Schnürer et al. , 2007 ) have showed the stability of Euclidean space under the Ricci flow for a small C0 perturbation . Koch et al . ( Koch & Lamm , 2012 ) have given the stability of the Euclidean space along with the Ricci flow in the L∞-Norm . Moreover , for the decay of the L∞-Norm on Euclidean space , Appleton ( Appleton , 2018 ) has given the proof of a different method . Considering the stability of integrable and closed Ricci-flat metrics , Sesum ( Sesum , 2006 ) has proved that the convergence rate is exponential because the spectrum of the Lichnerowicz operator is discrete . Furthermore , Deruelle et al . ( Deruelle & Kröncke , 2021 ) have proved that an asymptotically locally Euclidean Ricci-flat metric is dynamically stable under the Ricci flow , for an L2 ∩ L∞ perturbation on non-flat and non-compact Ricci-flat manifolds . If we embed a Riemannian n-dimensional manifold in the neural network , then we are training the neural network on the dynamic manifold . The most famous method for training neural networks on manifolds is the natural gradient ( Amari , 1998 ) . However , for the Riemannian manifold corresponding to the KL divergence ( representing the natural gradient ) , its stability and convergence are still unknown ( Martens , 2020 ) . As a way of manifold evolution , Ricci flow seems to be an excellent choice to ensure that neural networks are trained on dynamic and stable manifolds ( Glass et al. , 2020 ; Jejjala et al. , 2020 ) . But the research on the relationship between the two has not yet sprouted . In this paper , we consider a complete Riemannian n-dimensional manifold ( Mn , g ) endowed with linearly nearly Euclidean metrics g ( t ) = δ + γ ( t ) . First of all , we prove the stability of linearly nearly Euclidean manifolds under the Ricci-DeTurck flow in the L2-Norm if initial metrics are integrable and linearly stable , i.e . has a manifold structure of finite dimension . We mean that any Ricci-DeTurck flow which starts from near g exists for all time and converges to a linearly nearly Euclidean metric near g. Note that we use the Einstein summation convention and denote generic constants by C or C1 . Furthermore , we define and construct linearly nearly Euclidean manifolds based on information geometry and mirror descent algorithm . Based on a symmetrized convex function , we obtain the linearly nearly Euclidean divergence which is used to calculate the steepest descent gradient in linearly nearly Euclidean manifolds . Experimentally , when we use the approximated steepest descent gradient flow to learn several neural networks on classification tasks , we observe the evolution of its metric is consistent with the micro-surgery process under the Ricci-DeTurck flow . 2 RICCI FLOW . Let us introduce a partial differential equation , the Ricci flow , without explanation . The concept of the Ricci flow first published by Hamilton ( Hamilton et al. , 1982 ) on the manifold M of a timedependent Riemannian metric g ( t ) with the initial metric g0 : ∂ ∂t g ( t ) = −2 Ric ( g ( t ) ) g ( 0 ) = g0 ( 1 ) where Ric denotes the Ricci curvature tensor whose definition can be found in Appendix A . The purpose of the Ricci flow is to prove Thurston ’ s Geometrization Conjecture and Poincaré Conjecture because the Ricci flow is like a surgical scalpel , trimming irregular manifolds into regular manifolds to facilitate observation and discussion ( Sheridan & Rubinstein , 2006 ) . In general , in order to possess good geometric and topological properties , we expect the metric to become converge and round with the help of the Ricci flow . “ become round ” means that the solution will not shrink to a point but converge to a constant circle . However , in most cases , we do not even know the convergence of the solution and whether the solution will develop a singularity . Next , we will discuss these issues for brevity . 2.1 SHORT TIME EXISTENCE . To show that there exists a unique solution for a short time , we must check if the system of the Ricci flow is strongly parabolic . Theorem 1 When u : M× [ 0 , T ) → E is a time-dependent section of the vector bundle E whereM is some Riemannian manifold , if the system of the Ricci flow is strongly parabolic at u0 then there exists a solution on some time interval [ 0 , T ) , and the solution is unique for as long as it exists . Proof . The proofs can be found in ( Ladyzhenskaia et al. , 1988 ) . Definition 1 The Ricci flow is strongly parabolic if there exists δ > 0 such that for all covectors ϕ 6= 0 and all symmetric hij = ∂gij ( t ) ∂t 6= 0 , the principal symbol of −2 Ric satisfies [ −2 Ric ] ( ϕ ) ( h ) ijhij = gpq ( ϕpϕqhij + ϕiϕjhpq − ϕqϕihjp − ϕqϕjhip ) hij > δϕkϕkhrshrs . Since the inequality can not always be satisfied , the Ricci flow is not strongly parabolic , which makes us unable to prove the existence of the solution based on Theorem 1 . It is possible to understand which parts have an impact on its non-parabolic by the linearization of the Ricci curvature tensor . Lemma 1 The linearization of −2 Ric can be rewritten as D [ −2 Ric ] ( h ) ij = gpq∇p∇qhij +∇iVj +∇jVi +O ( hij ) where Vi = g pq ( 1 2 ∇ihpq −∇qhpi ) . ( 2 ) Proof . The proofs can be found in Appendix B.1 . The term O ( hij ) will have no contribution to the principal symbol of −2 Ric , so ignoring it will not affect our discussion of this problem . By carefully observing the above equation , one finds that the impact on the non-parabolic of the Ricci flow comes from the terms in V , not the term gpq∇p∇qhij . The solution is followed by the DeTurck Trick ( DeTurck , 1983 ) that has a timedependent reparameterization of the manifold : ∂ ∂t ḡ ( t ) = −2 Ric ( ḡ ( t ) ) − L ∂ϕ ( t ) ∂t ḡ ( t ) ḡ ( 0 ) = ḡ0 + d , ( 3 ) where d is a symmetric ( 0,2 ) -tensor on M. See Appendix B.2 for details . By choosing ∂ϕ ( t ) ∂t to cancel the effort of the terms in V , the reparameterized Ricci flow is strongly parabolic . Thus , one can say that the Ricci-DeTurck flow has a unique solution , the pullback metric , for a short time . 2.2 CURVATURE EXPLOSION AT SINGULARITY . In this subsection , we will present the behavior of the Ricci flow in finite time and show that the evolution of the curvature is close to divergence . The core demonstration is followed with Theorem 4 , which requires some other proof as a foreshadowing . Theorem 2 Given a smooth Riemannian metric g0 on a closed manifoldM , there exists a maximal time interval [ 0 , T ) such that a solution g ( t ) of the Ricci flow , with g ( 0 ) = g0 , exists and is smooth on [ 0 , T ) , and this solution is unique . Proof . The proofs can be found in ( Sheridan & Rubinstein , 2006 ) . Theorem 3 LetM be a closed manifold and g ( t ) a smooth time-dependent metric onM , defined for t ∈ [ 0 , T ) . If there exists a constant C < ∞ for all x ∈M such that∫ T 0 ∣∣∣∣ ∂∂tgx ( t ) ∣∣∣∣ g ( t ) dt ≤ C , ( 4 ) then the metrics g ( t ) converge uniformly as t approaches T to a continuous metric g ( T ) that is uniformly equivalent to g ( 0 ) and satisfies e−Cgx ( 0 ) ≤ gx ( T ) ≤ eCgx ( 0 ) . Proof . The proofs can be found in Appendix B.3 . Corollary 1 Let ( M , g ( t ) ) be a solution of the Ricci flow on a closed manifold . If |Rm |g ( t ) is bounded on a finite time [ 0 , T ) , then g ( t ) converges uniformly as t approaches T to a continuous metric g ( T ) which is uniformly equivalent to g ( 0 ) . Proof . The bound on |Rm |g ( t ) implies one on |Ric |g ( t ) . Based on Equation ( 1 ) , we can extend the bound on | ∂∂tg ( t ) |g ( t ) . Therefore , we obtain an integral of a bounded quantity over a finite interval is also bounded , by Theorem 3 . Theorem 4 If g0 is a smooth metric on a compact manifoldM , the Ricci flow with g ( 0 ) = g0 has a unique solution g ( t ) on a maximal time interval t ∈ [ 0 , T ) . If T < ∞ , then lim t→T ( sup x∈M |Rmx ( t ) | ) =∞ . ( 5 ) Proof . For a contradiction , we assume that |Rmx ( t ) | is bounded by a constant . It follows from Corollary 1 that the metrics g ( t ) converge uniformly in the norm induced by g ( t ) to a smooth metric g ( T ) . Based on Theorem 2 , it is possible to find a solution to the Ricci flow on t ∈ [ 0 , T ) because the smooth metric g ( T ) is uniformly equivalent to initial metric g ( 0 ) . Hence , one can extend the solution of the Ricci flow after the time point t = T , which is the result for continuous derivatives at t = T . This tell us that the time T of existence of the Ricci flow has not been maximal , which contradicts our assumption . In other words , |Rmx ( t ) | is unbounded . As approaching the singular time T , the Riemann curvature |Rm |g ( t ) becomes no longer convergent and tends to explode .

Summary: This paper investigates linearly nearly Euclidean metrics on Riemannian manifolds under the Ricci flow. Analysis, focusing on the stability and convergence of the evolution of such metrics, is presented. In addition, the utility of the analysis for measuring the approximation of gradient flow in the context of training neural networks is demonstrated.