Symmetric Spaces Rolling on Flat Spaces

The Journal of Geometric Analysis (2023) 33:94 https://doi.org/10.1007/s12220-022-01179-5 Symmetric Spaces Rolling on Flat Spaces V. Jurdjevic1 · I. Markina2 · F. Silva Leite3,4 Received: 30 September 2022 / Accepted: 19 December 2022 © The Author(s) 2023 Abstract The objective of the current paper is essentially twofold. Firstly, to make clear the difference between two notions of rolling a Riemannian manifold over another, using a language accessible to a wider audience, in particular to readers with interest in applications. Secondly, we concentrate on rolling an important class of Riemannian manifolds. In the first part of the paper, the relation between intrinsic and extrinsic rollings is explained in detail, while in the second part we address rollings of symmetric spaces on flat spaces and complement the theoretical results with illustrative examples. Keywords Semi-Riemannian manifolds · Group actions · Cartan decomposition · Intrinsic and extrinsic rolling · Stiefel manifolds Mathematics Subject Classification Primary: 53C35 · 53A35 · 53C50 · 53B21 · 53C25; Secondary: 53A17 · 53C17 1 Introduction In the contemporary literature, there exist two notions of rolling a Riemannian manifold over another, which more recently have also been extended to the semi-Riemannian case. One of these notions is intrinsic rolling, which does not require that the Riemannian manifolds are embedded. This concept uses the intrinsic geometry of the B I. Markina V. Jurdjevic F. Silva Leite 1 Department of Mathematics, University of Toronto, Toronto, ON M5S 3G3, Canada 2 Department of Mathematics, University of Bergen, P.O. Box 7803, 5020 Bergen, Norway 3 Institute of Systems and Robotics, University of Coimbra, Pólo II, 3030-290 Coimbra, Portugal 4 Department of Mathematics, University of Coimbra, Largo D. Dinis, 3000-143 Coimbra, Portugal 0123456789().: V,-vol 123 94 Page 2 of 33 V. Jurdjevic et al. manifolds only, and for Riemannian surfaces was introduced by Agrachev and Sachkov in [1] and by Bryant and Hsu in [2], and later studied for manifolds of higher dimensions, for instance, in [3, 4] and [8]. Extension to the semi-Riemannian situation appeared in [21]. Another definition of rolling initiated by Nomizu in [23] and presented more formally by Sharpe in [25] is the extrinsic rolling, which makes use of the isometric embedding of the manifolds in an ambient (semi)-Euclidean vector space V , so that the rolling is described in terms of the action of the group SE(V ) of oriented isometries of V . More recent works that use the extrinsic rolling are, for instance, [6, 11, 12, 16, 18, 20, 22, 26]. As far as we know, only in [8] and [21] both notions of rolling were addressed, the first for the Riemannian case and the second for the semi-Riemannian case. These two works are rather theoretical for researchers interested in applications of rolling motions but that do not have a strong background in differential geometry. The purpose of the current paper is essentially twofold. Firstly, we want to elucidate the difference between the two notions of rolling using a language that is more accessible to those with practical interest in rolling motions but less familiar with semi-Riemannian geometry. Secondly, we make transparent the relation between the geometry of the symmetric spaces and its rolling on flat spaces. Our main message is that the transitive action τ of a Lie group on a symmetric space completely defines its rolling along a chosen curve on the manifold. The differential map dτ is the isometry between the tangent spaces (after an identification of related vector spaces), that also matches the parallel vector fields on the rolling curves. Examples of semi-Riemannian symmetric spaces are provided, together with how to construct both types of rollings. It is always assumed throughout the paper that non-holonomic constraints of no-slip and no-twist are required in both situations, and the rollings are confined to semiRiemannian manifolds. The organization of the paper is the following. After setting the notation, we discuss in Sect. 3 the intrinsic rolling versus extrinsic. Section 4 is dedicated to rolling of symmetric spaces on flat spaces. Finally, we include Sect. 5 with the rolling of Stiefel manifolds, in order to illustrate the difference in the construction of rolling motions for a reductive homogeneous manifold, that is not a symmetric semi-Riemannian manifold. 2 Background and Notations In this section, we revisit the most important known concepts and results that will be used in the paper, and introduce the necessary notations. The main reference is the book of O’Neill [24], where the reader may find further details. 2.1 Semi-Riemannian Manifolds A semi-Riemannian manifold M is a smooth manifold endowed with a non-degenerate symmetric tensor g(. , .). We write n = dim M, and denote by p the number of positive eigenvalues of the tensor g, so that n − p is the number of negative eigenvalues of 123 Symmetric Spaces Rolling on Flat Spaces Page 3 of 33 94 g. The crucial example of a semi-Riemannian manifold is the semi-Euclidean vector space R p,n− p with the semi-Euclidean product x, y p,n− p = p  xk yk − k=1 n  xk yk , x, y ∈ R p,n− p . k= p+1 Another important example is a vector space V with a bilinear symmetric nondegenerate form (. , .) p,n− p . We will often refer to (. , .) p,n− p as a scalar product and simply write (. , .) in case there is no need to specify the signature. Let (V , (. , .) p,n− p ) be a semi-Riemannian vector space. The isometric embedding map will be denoted by ι: M → V . On existence of such an embedding see [5]. For the moment, we will identify the manifold M with its image under the embedding, that is, notationally, ι(M) = M. The semi-Riemannian metric g(. , .) on the embedded manifold M is inherited from the semi-Riemannian product (. , .) p,n− p in the ambient space V . The isometric embedding of M into V splits the tangent space of V , at a point m ∈ M, into a direct sum: (1) Tm V = Tm M ⊕ (Tm M)⊥ , m ∈ M, where ⊥ denotes the orthogonal complement with respect to (. , .) p,n− p . Note that the tangent space Tm M and the normal space (Tm M)⊥ are non-degenerate subspaces of (V , (. , .) p,n− p ). According to this, any vector v ∈ Tm V , m ∈ M can be written uniquely as the sum v = v  + v ⊥ , where v  ∈ Tm M, v ⊥ ∈ (Tm M)⊥ . In what follows, ∇ denotes the Levi-Civita connection on the ambient space V , and ∇ for the Levi-Civita connection on M. If X and Y are tangent vector fields on M, and ϒ is a normal vector field on M, then ⊥    ∇ X Y ( p) = ∇ X̄ Ȳ ( p) , ∇ X⊥ ϒ( p) = ∇ X̄ ϒ̄( p) , p ∈ M, where X̄ , Ȳ , and ϒ̄ are any local extensions to V of the vector fields X , Y , and ϒ, Dα(t) Z (t) respectively. If Z (t) and ϒ(t) are vector fields along a curve α(t), we use dt D⊥ α(t) ϒ(t) for the normal to denote the covariant derivative of Z (t) along α(t) and dt covariant derivative of ϒ(t) along (...truncated)