Geodesic completeness in general relativity There are well-known definitions of complete geodesic and geodesically complete spacetime:

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*A geodesic is complete if an affine parameter for the geodesic extends to ±∞.


*A spacetime is geodesically complete if all inextendible causal geodesics are complete.
For example, Minkowski spacetime is geodesically complete, as is the spacetime describing a static spherical star.
However, the Kruskal spacetime (extension of Schwarzschild spacetime) is geodesically incomplete because some geodesics have $r \to 0$ in finite affine parameter and hence cannot be extended to infinite affine parameter.
Is this notion usefull in general relativity?
What is physical interpretation of geodesical completeness?
How are geodesically complete and incomplete spaces different in physical sence?
Is it possible to extend incomplete spacetime to geodesically complete?
Edit: useful link Light Rays, Singularities, and All That
 A: 
Is this notion usefull in general relativity?

Yes.

What is physical interpretation of geodesical completeness?

Typically this means there is a singularity in the spacetime that prevents the geodesics from being continued, for example at the center of a black hole, or the Big Bang singularity. It could also mean there is some boundary to the spacetime, or there is some puncture (for example if you remove the point at the origin).
However the physical interpretation of a singularity itself is complicated. Most physicists believe that Nature does not have singularities, and the singularities in the classical solutions of GR are really a sign that GR is breaking down. So geodesic incompleteness may say more about the incompleteness of GR, than of spacetime.

How are geodesically complete and incomplete spaces different in physical sence?

See above.

Is it possible to extend incomplete spacetime to geodesically complete?

Not without changing the spacetime. For example, if the spacetime has a puncture you could "fill in" the puncture. But you wouldn't be able to extend the Scwarzschild solution in a way that avoids the singularity.

By the way, a note on terminology. You said:

Kruskal spacetime (extension of Schwarzschild spacetime)

A more common usage of these words would be to say there is a black hole spacetime (the Schwarzschild solution), which can be described in Schwarzschild coordinates or Kruskal coordinates. The word "spacetime" should refer to the invariant geometry; what is different between Kruskal and Schwarzschild are the coordinates, not the spacetime itself.
A: 
Is this notion usefull in general relativity?

In GR, more useful than geodesic completness is geodesical maximality, which tells you that every geodesic either goes to infinite values of affine parameter or ends in singularity. That is due to the fact, that GR spacetimes can have singularities, so demanding geodesical completness is too restrictive.
However you want to to extend the manifold so that it is maximal. The spacetime where some geodesics would end at finite amount of affine parameter without arriving at singularity is quite a weird idea. Imagine taking a ship and traveling on one of this geodesics. This would mean that you would reach the end of the geodesics in finite amount of proper time. Then what? You cease to exist without any good reason?
