Leakage inductance is function of the air gap
I would like to know why leakage inductance is function of the air gap
  transformer. How does it influence the leakage flux ? What is the
  relation between them ?


With a transformer core of high effective magnetic permeability, you could say that the flux produced by one winding 99.9% couples to the other winding. The closer the coupling is to 100%, the smaller is the leakage flux. 

For a core with moderate effective permeability, you might attribute a coupling factor of only 99%. For a core of low effective permeability you might "say" the coupling is around 90%.

Adding a gap to a core of high permeability makes the effective permeability of a transformer smaller and, in turn, the coupling also reduces. A bigger gap reduces the effective permeability even more.

In other words, it isn't the gap itself that creates all the leakage flux but rather the drop in overall permeability making the core "less attractive" of a "path" for magnetic fields to flow along.

An effective relative permeability of 10 is only 10 times that of air and, although most flux would flow through that core, a decent proportion (say 10%) will short cut the core and return back through the air. Whether you use a high permeability core with a gap to bring it down from 1000 to 10 or, you use an un-gapped core of low permeability (10), the effect is the same.