Renormalized differential geometry on path space: Structural equation, curvature
The structure of manifolds of maps of a compact manifold attracts the interest of mathematicians for more than thirty years starting from the works by Ebin, Marsden, Eliasson and others [see for example \textit{D. G. Ebin} and \textit{J. Marsden}, Ann. Math., II. Ser. 92, 102-163 (1970; Zbl 0211.57401), \textit{H. I. Eliasson}, J. Differ. Geom. 1, 169-194 (1967; Zbl 0163.43901)]. Those manifolds are infinite-dimensional and should be carefully studied to avoid divergence and explosions of some classical differential geometry structures. The authors describe differential geometry of a manifold \(P_{m_0}(M)= \{p(\cdot)\in C([0,1], M):p(0)=m_0\} \) where \(M\) is a compact Riemannian manifold with a model space \(R^d.\) As far as \(P_{m_0}(M)\) comes to be a probability space being endowed with the natural filtration under Wiener measure induced by the Brownian motion \(p\) on \(M\) they use such a powerful tool as Malliavin calculus to describe its differential geometry structures. In particular such notions as connection and especially Levi-Civita connection, parallel translation, curvature tensor, structure equations and others are constructed for the infinite-dimensional manifold \(P_{m_0}(M)\). The main tool the authors develop and work with is the stochastic Stratonovich integral on \(P_{m_0}(M)\) and stochastic differential equation (SDE) theory associated with it. The Stratonovich integral for a vector field \(Z\) on \(M\) is determined as a limit of the Riemannian sum \[ \frac{1}{2}\sum (Z_{p(\sigma)},[\text{exp}^{-1}_{p(\sigma)}(p(\sigma^+)) - \text{exp}^{-1}_{p(\sigma)}(p(\sigma^-))])_{T_{p(\sigma)}(M)} \] where exp is an exponetial map on \(M\), \(p(t)\in M\) is a diffusion process having the Beltrami-Laplace operator \(\Delta_M\) for its generator, \(S\) is a finite part of \([0,1]\) and \[ \sigma^+=\text{inf}\{\sigma'\in S\cup 0:\sigma\leq \sigma'\},\quad \sigma^-=\text{sup}\{\sigma'\in S\cup 0:\sigma'\leq \sigma\}. \] To determine and investigate the necessary differential geometry constructions on \(P_{m_0}(M)\) the authors start with corresponding finite-dimensional objects determined for chosen \(S\) and then give the conditions to ensure the validity of a limiting procedure. The important feature of the exposition is the application of moving frames and the work in the context of the frame bundle over \(M\) which allows to make computations in canonical coordinates. The approach is developed in the framework of infinite-dimensional theory of anticipative stochastic integrals which plays an important part in deriving the results. Dealing with infinite-dimensional differential geometry objects the authors introduce the hypoelliptic renormalization and show that some of them do not admit it. It leads to the notion of a tangent process which is carefully studied in the paper. The paper shows how fruitful could be the application of SDE theory to infinite-dimensional analysis. It should be noticed that a bit different approach to differential geometry constructions for infinite-dimensional manifolds and some other applications of SDE to analysis the reader could find in the book ``Stochastic equations and differential geometry'' (1990; Zbl 0696.60053) by the reviewer and \textit{Yu L. Daletskij}.
scientific article; zbMATH DE number 944393