arXiv:2202.02622v1 [math.DG] 5 Feb 2022 Killing forms on Riemannian Poisson warped product space Buddhadev Pal* and Pankaj Kumar 1 Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-221005, India. *Email Id: pal.buddha@gmail.com Email Id: pankaj.kumar14@bhu.ac.in Abstract A formal treatment of Killing 1-form and 2-Killing 1-form on Riemannian Poisson manifold, Riemannian Poisson warped product space are presented. In this way, we obtain Bochner type result on compact Riemannian Poisson manifold, com- pact Riemannian Poisson warped product space for Killing 1-form and 2-Killing 1-form. Finally, we give the characterization of a 2-Killing 1-form on (R 2 ,g, Π). Key words : Warped product, Killing 1-forms, Levi-Civita contravariant con- nection, Poisson structure, Riemannian Poisson manifold. 1. Introduction To provide the example of Riemannian spaces having negative curvature Bishop and O’Neill [1] introduced the notion of warped space. From then on original and generalized form of warped product spaces have been widely discussed by both mathematicians and physicists [2, 3, 4, 5, 6, 7, 8, 9]. Let ( ˜ M 1 , ˜ g 1 ) and ( ˜ M 2 , ˜ g 2 ) are two pseudo-Riemannian manifolds with positive smooth function f on ˜ M 1 . Let π 1 : ˜ M 1 × ˜ M 2 → ˜ M 1 and π 2 : ˜ M 1 × ˜ M 2 → ˜ M 1 are the projections. The warped product ˜ M = ˜ M 1 × f ˜ M 2 is the product manifold ˜ M 1 × ˜ M 2 endowed with the metric tensor ˜ g = π * 1 (˜ g 1 )+(f ◦ π 1 ) 2 π * 2 (˜ g 2 ), called warped product and ordered-pair ( ˜ M, ˜ g) known as warped product space. Here ˜ M 1 , ˜ M 2 and f are respectively known as base space, fiber space and warping function of the warped product space ( ˜ M, ˜ g) and * stand for pull-back operator. Killing vector fields are the relevant object for the geometry specially in pseudo- Riemannian geometry where mathematicians characterized the existence of Killing vector fields. Killing vector fields are also studied by many physicists in the prospective of general relativity in which these are expounded in the term of sym- metry. Bochner [10, 11, 12], studied in detail Killing vector fields and provided various remarkable results. K. Yano [13, 14], consider a compact orientable Rie- mannian spaces with boundary and generalized the Bochner technique to study Killing vector fields on it. S. Yorozu [15, 16], discussed the non-existence of Killing vector fields on complete Riemannian spaces and also did the same for 1 Second author’s work is funded by UGC, India in the form of JRF [1269/(SC)(CSIR-UGC NET DEC. 2016)]. 1