I just started reading about Calabi-Yau manifolds and most of the sources I came across defined Calabi-Yau manifold in a different way. I can see that some of them are just same and I can derive one from other. But my question is the following :


"What is the most strict definition of Calabi-Yau Manifolds" 


By that I mean the definition from which all the others follow. 



Calabi - Yau Manifolds
There are several different "definitions" of Calabi-Yau manifolds, not all equivalent, and not all contained in one general definition. A good discussion of some of these inequivalent definitions can be found in Joyce's book:


[Compact Manifolds with Special Holonomy](http://books.google.ca/books?id=c3P-YUD8GZQC&dq=joyce+compact+manifolds&hl=en&ei=417ATLScGZSpnQeR5aWJCg&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDEQ6AEwAA)


The other answers you have gotten so far seem to be from the algebraic geometry side of things, and are fine in that context. From a Riemannian geometry point of view, the most natural definition of a Calabi-Yau manifold (whether compact or non-compact) is a $2n$-dimensional Riemannian manifold for which the holonomy of the Levi-Civita connection is exactly $SU(n)$. Allowing the holonomy to be a proper subgroup of $SU(n)$ is also common. In that case, hyperKahler (which is holonomy $Sp(n/2)$ in dimension $4n$) can also be considered as being Calabi-Yau, for example.


This Riemannian geometry definition is equivalent to the existence of the "Calabi-Yau package": a Riemannian metric $g$, an integrable complex structure $J$ (orthogonal with respect to $g$) together which induce the associated Kahler form $\omega$ by $\omega(X,Y) = g(JX, Y)$, and a *holomorphic volume form* $\Omega$, which is a holomorphic $(n,0)$-form on $M$. These tensors must satisfy:


(1) $\nabla \omega = 0$ (equivalent to $\nabla J = 0$, the Kahler condition) This is also equivalent to $d\omega = 0$ because we are assuming $J$ to be integrable.


(2) $\nabla \Omega = 0$


(3) $\frac{\omega^n}{n!} = c\_n \, \Omega \wedge \bar \Omega$ for some universal *constant* $c\_n$ depending only on the dimension which I can never remember.


These conditions imply, in particular that $g$ is Ricci-flat and $c\_1(M) = 0$. Also, if the holonomy is exactly $SU(n)$ rather than a proper subgroup, then it also follows that $h^{p,0} = h^{0,p} = 0$ for all $1 \leq p \leq n-1$.


