Counting geodesics on compact symmetric spaces (Q6544453)

For a Riemannian manifold \((M,g)\) the Riemannian exponential map \(\exp_p\colon T_p M\to M\) is defined by assigning to a vector \(v\in T_p M\) the value at time \(t = 1\) of the unique geodesic \(\gamma\colon \mathbb{R}\to M\) defined by \(\gamma'(0) = v\). Hence, if \(q\in M\) is a given point, then understanding the preimage \(\mathrm{exp}_p^{-1}(q)\subseteq T_p M\) is equivalent to understanding all geodesics in \(M\) linking \(p\) and \(q\). The problem of determining this preimage is called the problem of counting geodesics by the authors of the present article.\N\NThe authors study this problem explicitly on compact symmetric spaces by using the Lie-theoretic description of a compact symmetric space. In particular we remark that the authors study arbitrary compact symmetric spaces, i.e., their study is not just restricted to symmetric spaces of compact type or to simply connected symmetric spaces.\N\NRecall that every symmetric space \(M\) is a homogeneous space \(M = U/K\) for a compact Lie group \(U\) and a closed subgroup \(K\). As basepoint one takes the point \(p_0 = [e]\in U/K\). Choose another point \(q\in M\) and a tangent vector \(v\in T_{p_0}M\) with \(\exp_{p_0}(v) =q\). Note that if \(k\in K\) fixes \(q\), then \(\mathrm{exp}_{p_0}(k.v) = q \), since \(K\) acts isometrically on \(U/K\). Consequently, the group \N\[\NK^q = \{ k\in K\,|\, k \text{ fixes }q\} \N\]\Nacts on the preimage \(\mathrm{exp}_{p_0}^{-1}(q)\). Denote the orbit of \(v\) under this action by \(\mathcal{F}(v)\). If \(r \) is the rank of the symmetric space \(M\) then there is a maximal abelian subspace \(\mathfrak{t}\subseteq T_{p_0} M\), with \(\mathrm{dim}(\mathfrak{t}) = r\). This maximal abelian subspace is the universal covering of a flat torus and there is a corresponding lattice \(\Gamma\subseteq \mathfrak{t}\). The main result of the article is the following.\N\NTheorem A. Let \(M = U/K\) be a compact Riemannian symmetric space and let \(v\in T_{p_0}M\) be a tangent vector with \(\mathrm{exp}_{p_0}(v) = q\). The preimage of \(q\) under \(\mathrm{exp}_{p_0}\colon T_{p_0} M\to M\) is given by the union of the orbits \N\[\N\exp_{p_0}^{-1}(q) = \bigcup_{w\in\Gamma} \mathcal{F}( v +w ).\N\]\NThis union is disjoint, each connected component of \(\mathcal{F}(v+w)\) corresponds to a homotopy class of geodesics joining \(p\) and \(q\) and the dimension of \(\mathcal{F}(v+w)\) as well as the number of connected components can be read off from the Lie-theoretic data of \(M\). Moreover, all connected components of \(\mathcal{F}(v+w)\) are diffeomorphic to each other.\N\NThe authors prove this theorem by a very detailed analysis of the Lie-theoretic description of a compact symmetric space \(M\). The authors use the main result to determine the fundamental group of a compact symmetric space.\N\NTheorem B. Let \(M\) be a compact symmetric space, let \(\Gamma\) be the lattice induced by the maximal torus as above and let \(\Gamma_0\subseteq \Gamma\) be the fundamental lattice. Then there is an isomorphism \(\pi_1 (M) \cong \Gamma/\Gamma_0\).\N\NSee Section 2.4 of the article for the notion of fundamental lattice. This behaviour of the fundamental group has been well-known for symmetric spaces of compact type.\N\NThe main result of the article (Theorem A) is then also used to obtain new and simplified proofs of classical results about the cut locus as well as the conjugate locus of compact symmetric spaces by \textit{R. Crittenden} [Can. J. Math. 14, 320--328 (1962; Zbl 0105.34801)] and \textit{T. Sakai} [Hokkaido Math. J. 6, 136--161 (1977; Zbl 0361.53048)].