Symmetries and locally s-regular manifolds
Any locally s-regular Riemannian manifold admits a particular (1,1) tensor field S called a regular symmetry tensor field, which reduces to - I for the special case of a symmetric space. The authors assume that a given Riemannian manifold (M,g) admits such a tensor field S and consider the corresponding local geodesic symmetries \(s_ m\), \(m\in M\), defined by \(s_ m=\exp _ m\circ S_ m\circ \exp _ m^{-1}\). They obtain four sets of necessary and sufficient geometric conditions under each of which (M,g) is locally s-regular with respect to S and the family \(\{s_ m: m\in M\}.\) The first of these is that, for each \(m\in M\), \(s_ m\) should preserve the normal volume function on all sufficiently small geodesic disks centred at m. The next two characterizations require the s-invariance of the shape operator on geodesic spheres. Finally, it is well known that any Riemannian locally s-regular manifold carries a canonical s-invariant f-structure. Thus f is an s-invariant (1,1) tensor field such that \(f^ 3+f=0\) and \(g(fX,Y)+g(X,fY)=0\). Clearly the tensor fields h, k defined by \(h(X,Y)=g(X,fY)\) and \(k(X,Y)=g(X,(F^ 2+I)Y)\) are also s-invariant. The authors show that, conversely, if S is a regular symmetry tensor field on (M,g,f) such that for each \(m\in M\) the tensor fields h and k are locally \(s_ m\)-invariant then (M,g) is locally s-regular with symmetry tensor field S. This extends their earlier characterization (to appear) for the case when S has odd order and f is, necessarily, almost Hermitian.
scientific article; zbMATH DE number 3971667
scientific article