Maximal totally geodesic submanifolds and index of symmetric spaces (Q342691)

The authors are concerned with the question of determining the index \(i(M)\) of an irreducible Riemannian symmetric space \(M\) of noncompact type (the index of \(M\) is the minimal codimension of the totally geodesic submanifolds of \(M\)).NEWLINENEWLINEThis paper is a continuation of the work of \textit{A. L. Onishchik} who introduced the concept of index and of the authors in the paper [``On the index of symmetric spaces'', Preprint, \url{arXiv:1401.3585}].NEWLINENEWLINEThe first main theorem of the paper is a characterization (by equivalent criteria and by a complete list) of the non-semisimple complete totally geodesic submanifold of the irreducible Riemannian symmetric space of noncompact type \(M\). In order to get to that result, the authors give a brief introduction to symmetric spaces and explain the notion of reflective submanifolds (such a manifold is a connected component of the fixed point set of an involutive isometry of \(M\)). In the words of the authors, ``[the] totally geodesic submanifolds of sufficiently small codimension in irreducible Riemannian symmetric spaces are reflective.'' Another concept introduced is that of \(R\)-space: the orbit \(K\cdot v=\text{Ad}(K)\,v\) when it is an irreducible symmetric space.NEWLINENEWLINEIt is known that the index is greater than or equal to the rank. The other main theorem stipulates when the rank and the index of an irreducible Riemannian symmetric space of noncompact type are equal (a complete list is provided). The result is proved in two parts; one section is devoted to showing that the spaces in the list indeed satisfy the condition and the next one to the proof that they are the only ones.NEWLINENEWLINEThe authors prove several additional results. They introduce the index \(i_r(M)\) which is the minimal codimension of the reflective submanifolds of \(M\) (so that \(i_r(M)\geq i(M)\)). In addition to some general criteria about \(i(M)\) involving \(i_r(M)\), they give complete lists of the irreducible Riemannian symmetric spaces of noncompact type \(M\) with \(i(M)=k\) for \(k=4\), 5, 6 (the cases \(i(M)\leq 3\) had been proved in their previous paper).NEWLINENEWLINEThey conclude their paper with an interesting conjecture: that \(i(M)=i_r(M)\) is always true except when \(M=G_2^2/\mathrm{SO}_4\).