Is curvature space-time has impact on the object geometry When we have e.g. metallic cube of dimensions 1x1x1m and we put it on the space 
without gravitational force the cube has equal 1x1x1m and we can use Euclidean geometry.


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*But when this cube move on the Earth's surface where the space-time is curvature and twisted this cube has not-Euclidean geometry? So now it not has equal 1m sides, and it volume is change?

*If this cube put on the surface of the earth and I use the laser distance meter to measuring the width of a edge on the base and the top of this cube I get different lengths because at the bottom the time runs slower than 1m above. So this cube is has realy more width in the bottom or it is just time dilation effect?

 A: Yes, in principle you can observe difference of size and form in view of the geodesic deviation. If you have a pair of particles of matter very close to each other, at rest in a reference frame, in a generic region of spacetime where curvature is present, they experience relative acceleration in view of the so-called geodesic deviation which is due to the fact that Riemann tensor is not zero. This implies that if you want to keep the particles at fixed relative distance you should apply forces. In a extended body these forces are called internal stresses. Internal stresses are related to the deformation of the object. Therefore a metallic cube must change its size and its shape may change in the presence of curvature in order to cancel the relative acceleration of its molecules, imposed by the "gravitational field". 
The second question is not well posed because your statement "at the bottom the time runs slower than 1m above" must be interpreted...
What you can compare is some interval of time referred to some Killing time used, for example, to define the notion of thermodynamic equilibrium. Interval of Killing time are identical everywhere, for instance the temporal period of electromagnetic wave emitted from the top to the bottom  are the same at the bottom and at the top.  However the measures of these intervals referred to ideal clocks staying respectively at the top and at the bottom turn out to be different. 
However, if our measurement are all performed staying at the top or staying at the bottom, no difference of interval of time are revealed. 
Moreover, the (constant even in GR) value of the speed of the light is always referred to the proper time and not the Killing time, and the notion of length is coherent with this choice.
So the answer to your second question is NO, that is not the way to observe a geometric deformation.
A: First of all, the curvature of space is expected to change (in proportion) everything you use to measure the cube. Including scales, lasers etc. everything. And so, you would not observe any difference. 
However, the curving of space is not necessarily a geometrical one. Even if it is geometrical, It may not necessarily be along the directions in which we observe its effects.
We see light bending due to curved space. That does not mean space is bent in the same direction. 
For example, a lens also bends light, but we know that lens itself is not bent in same direction as it bends light into. Actually, it is bent in a perpendicular direction. Not a very good analogy, but just to give an idea.
Therefore, I would say, that while the mathematics works beyond any reasonable doubt, the mechanism, and shape/nature of curving itself is not known/described. 
However, I guess, the mathematics does consider the curving as geometrical and in same directions as the observed effects. And the math works very elegantly.
