I am obviously not familiar with differential geometry. But some times I do want to know detailed answers to the following questions? May someone help?

When will there be a longest simple closed geodesic on a metric space? Of course, this is too general a question. To be more touchable, what is the case for Riemannian manifolds with non-positive curvature?

Or more generally is there any reference for the the relationship between longest/shortest simple closed curve, diameter, area and curvature?

Growth of the number of simple closed geodesics on hyperbolic surfaces, Annals of Mathematics,168(2008), 97–125, which gives an estimate for how fast the number of such simple closed geodesics grows with given length $L$. Boundedness not only fails, but fails spectacularly on a hyperbolic surface of genus $g>1$. – Robert Bryant Jan 22 '13 at 17:27