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In the present paper we examine a projectively flat spacetime solution of$F(R)$-gravity theory. It is seen that once we deploy projective flatness inthe geometry of the spacetime, the matter field has constant energy density andisotropic pressure. We then make the condition weaker and discuss the effectsof projectively harmonic spacetime geometry in $F(R)$-gravity theory and showthat the spacetime in this case reduces to a generalised Robertson-Walkerspacetime with a shear, vorticity, acceleration free perfect fluid with aspecific form of expansion scalar presented in terms of the scale factor. Roleof conharmonic curvature tensor in the spacetime geometry is also brieflydiscussed. Some analysis of the obtained results are conducted in terms ofcouple of $F(R)$-gravity models.

How a projectively flat geometry regulates $F(R)$-gravity theory?