“Harmonic Forms and Killing Tensor Fields on Almost Grayan Manifold”

International Journal of Research and Scientific Innovation (IJRSI) | Volume VI, Issue VIII, August 2019 | ISSN 2321–2705

“Harmonic Forms and Killing Tensor Fields on Almost Grayan Manifold”

Dr. Banke Bihari, Dr. S.P Pandey

Department of Mathematics, R.B.S. College, Agra, India

Abstract:-In this paper we have studied different aspect of para complex and almost para complex manifold which is similar to almost Grayan manifold.

I. INTRODUCTION

An almost para complex structure F is intregrable if an only if N_F=0.

Proof:
Consider the two projections π±:Tm -> T±M,

π±:=1/2(Fd±F)

Then by the Frobeneus theorem, the integrability of T+M and T-M is equivalent to respectively

π-[π+X, π+Y]  =0 and 
π+[π-X, π-Y]  =0
For all vector fields X and Y. The sum and the difference of these expressions are proportional to N_F(X,Y) and FN_F(X,Y)
Ex.1 Any para complex vector space (V,F) can be considered as a Para complex manifold, with constant Para-complex structure.

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