Geodesic flows on path spaces
type: Item
instance of: scholarly article
title: Geodesic flows on path spaces
zbMATH Open document ID: 1025.58016
DOI: 10.1007/BF02881883
author: Yong Liu
author: Kai-Nan Xiang
published in: Science in China. Series A
publication date: 2002-08-15T00:00:00Z
review text: Let \(m_0\) be a point in a compact, \(d\)-dimensional Riemannian manifold \(M\), and let \(P_{m_0}(M)\) be the set of continuous curves \(\gamma: [0,1]\to M\) such that \(\gamma(0)= m_0\). On the space \(P_{m_0}(M)\) one can define a measure \(\mu\) induced from the classical Wiener measure \(\mu_0\) on \(P_0(\mathbb{R}^d)\). For \(p\in P_{m_0}(M)\) one defines the tangent space \(T_p(P_{m_0}(M))= \{Z^0: Z(t)= t^p_{t\leftarrow 0}(z(t)): z\in H\}\). Here \(H\) denotes the Cameron-Martin space of curves in \(P_0(\mathbb{R}^d)\) with finite energy, and \(t^p_{t\leftarrow 0}\) denotes the parallel transport defined by Itô along the Brownian curve on \(M\) from \(p(0)\) to \(p(t)\). Cruzeiro and Malliavin defined a certain Markov linear connection \(\nabla\) on \(P_{m_0}(M)\), and one may also define a negative Markov connection in a very simlilar way. If one considers the geodesic flow transformations \(\{g^t\}\) of either of these connections, then one may ask if the measure \(\mu\) on \(P_{m_0}(M)\) is quasi-invariant under \(\{g^t\}\). For the Markov connection an affirmative answer to this question was obtained by X. D. Li. In the present article the authors obtain an affirmative answer for the negative Markov connection.
Mathematics Subject Classification ID: 58J65
Mathematics Subject Classification ID: 58B20
zbMATH DE Number: 1782103
zbMATH Keywords: path spaces
zbMATH Keywords: Wiener measure
zbMATH Keywords: quasi-invariance
zbMATH Keywords: geodesic flow
zbMATH Keywords: negative Markov connection
reviewed by: Patrick Eberlein
MaRDI profile type: Publication
cites work: Renormalized differential geometry on path space: Structural equation, curvature
cites work: A Cameron-Martin type quasi-invariance theorem for Brownian motion on a compact Riemannian manifold
cites work: Towards a Riemannian geometry on the path space over a Riemannian manifold
cites work: Quasi-invariance of the Wiener measure on the path space over a compact Riemannian manifold
cites work: A class of vector fields on path spaces
cites work: Existence and uniqueness of geodesics on path spaces
full work available at URL: https://doi.org/10.1007/bf02881883
OpenAlex ID: W1543895892
description: scientific article; zbMATH DE number 1782103