Frame bundle of Riemannian path space and Ricci tensor in adapted differential geometry
Let \({\mathcal H}\) be the Hilbert space \({\mathcal H}= \mathbb{H}^1([0, 1],\mathbb{R}^d)\) equipped with a continuous family of projectors \(\Pi_\sigma\) \((\sigma\in [0,1])\) and \(V\) a \(d\)-dimensional Riemannian manifold. The projection \(\Pi_\sigma\) \((\sigma\in [0,1])\) is defined by \(d_\tau(\Pi_\sigma z)= 1_{[0,\sigma]}(\tau) z(\tau)\). Further, the set \(P_{m_0}(V)\) of continuous maps from \([0,1]\) to \(V\) such that \(p(0)= m_0\) (\(m_0\) is a fixed point of \(V\)) is called the Riemannian path space of \(V\). With its natural filtration, under the Wiener measure induced by the Brownian motion of \(V\) associated to \(({1\over 2})\Delta\) (\(\Delta\) denotes the Laplace-Beltrami operator of \(V\)), the space \(P_{m_0}\) becomes the basic probability space. An isometric surjective map from \({\mathcal H}\) onto the Cameron-Martin tangent space \(T^1_p(P_{m_0}(V))\) is called a frame at the point \(p\in P_{m_0}(V)\), and especially the frame at \(p\) defined by the Itô parallel transport is called the canonic one. An adapted frame is a frame which interwings the projection operators \(\Pi_\lambda\) on \(T^1_p(P_{m_0}(V))\). The collection \(O(P_{m_0}(V))\) of all adapted frames is called the frame bundle. The projection operator \(\Pi_\lambda\) is defined by \((\Pi_\lambda(Z))(\tau)= Z_\tau\), for any \(\tau< \lambda\) and \((\Pi_\lambda Z)(\tau)= t^p_{\tau\leftarrow\lambda}(Z_\lambda)\), for any \(\tau> \lambda\), where \(Z_\tau= t^p_{\tau\leftarrow 0}(z(0))\), \(t^p_{\tau\leftarrow 0}\) denoting the Itô parallel transport. In the present paper, the authors proceed the Ricci calculus on the (Riemannian) path space \(P_{m_0}(V)\) under the Markovian connection developed in their previous paper [ibid. 139, No. 1, 119-181 (1996; Zbl 0869.60060)]. In the authors' setting, the renormalization means the restriction of the classical identities, as the Weitzenböck formula, to adapted vector fields. The main result of the present paper is that the renormalized Shigekawa-Ricci tensor associated to the Markovian connection of Riemannian path space over Ricci-flat Riemannian manifold vanishes.
scientific article; zbMATH DE number 1542421