Killing vectors of FLRW metric (in comoving coordinates) and zero modes of the scalar Laplacian

Based on an examination of the solutions to the Killing Vector equations for FLRW-metric in co moving coordinates, it is conjectured and proved that components (in these coordinates) of Killing Vectors, when suitably scaled functions, are modes of the corresponding. The complete such set of zero modes (infinitely many) are explicitly for the two-sphere. They are parametrised by an integer n. Forn\\, \\\, 2, all the solutions are (in the sense that they are well defined everywhere nor are -integrable). The 2-d vectors are also normalisable. The n=0 solutions constants (these correspond to the zero angular momentum solutions) are and normalizable. Not all of the n=1 solutions are regular but the vectors are normalizable. Of course, the action of scalar Laplacian independent significance only when acting on scalars. However, our have an unambiguous meaning as long as one works in this coordinate. As an intermediate step, the covariant Laplacians (vector Laplacians) of vectors are worked out for four-manifolds in two different ways, both which have the novelty of not explicitly needing the connections. It is shown that for certain maximally symmetric sub-manifolds (hypersurfaces one or more constant comoving coordinates) of the FLRW-spaces also, the Killing vector components are zero modes of their corresponding scalar. The Killing vectors for the maximally symmetric four-manifolds are out using the elegant embedding formalism originally due to\\"odinger. Some consequences of our results are worked out. Relevance to very recent works on zero modes in AdS/CFT correspondences, as well as on scenarios is briefly commented upon.