Genady

Right. The only issue is that I don't have any idea what we are discussing now and why. Yes, cylinder is flat. A cone is flat everywhere except the very tip. A bagel is not flat except along the very top and the very bottom. Etc.

Agreed. This detour was just a little exchange between @sethoflagos and me and should've been over with the @Markus Hanke's post above. However, some posters have picked up some comments as if they were standalone statements and have used them to educate the audience in an unrelated material.

You are correct. The discussion here is in terms of Riemann curvature, which is relevant for discussions on GR, usually. You are correct again. The mathematical treatment can be found in textbooks on differential geometry that include discussion on the spacetime of GR. Also, in bits and pieces, in articles on Internet. PS. The Riemann curvature is of course tensor rather than a matrix.

Yes, size and time. I think, this is the correct answer. I don't know. I don't know that theory well enough. To my very limited understanding, no. My answer to this is, Yes.

If the scale factor in FLRW metric is a constant, the space cannot expand or contract.

Yes, so? It is not a spacetime. The metric signature is different.

It goes like this. For the 3+1 dimensional spacetime to be flat, i.e., to have a vanishing curvature it has to be Minkowski, i.e., to allow coordinates in which the metric is Minkowski. For a spacetime with FLRW metric to allow this, the scale factor needs to be constant. Thus, with a non-constant scale factor, the spacetime with FLRW metric is not flat. We can keep one dimension, namely time, constant and consider the 3D slice separately. This 3D space can be Euclidean, hence flat.

I think that the issues has been already resolved, multiple times, e.g.,

The point of the above discussion was that even with k=0 precisely, the spacetime is not flat although the space is flat.

I am not sure about the smallest Plank; I've asked some questions about it for clarification in another thread but have never got a reply. Regardless, even in one splitting moment infinite number of universes may appear.

Yes, and I understand (although disagree) that there is. Moreover, some divisions create infinite numbers of the universes at one splitting point.

Yes, but there is infinite number of points on that finite line where the universes split.

A lack of definition is not a 'big problem.' Most of the concepts we use in life, science and even many in math lack definitions. For another example, AI recognizes faces, cars, molecules, etc. without definitions. Only when we understand something really well, we can come up with a good definition for it.

No, we are not.

Yes, it is extremely close to flat spatially, but the discussion here is about flatness of spacetime, and the spacetime of our universe is not flat.

Because expanding (or contracting) spacetime is not a flat spacetime.

A flat spacetime would be Minkowski spacetime. A homogenous isotropic universe is not.

On these scales the universe seems pretty flat spatially, as I understand, and this has no relation to the time translation symmetry, does it?

Thanks. I think I understand you.

Could you describe a bit of what you mean by this, please?

Einstein was invited to a dinner where he was asked by a hostess to "be so kind as to explain to my guests in a few words, just what is relativity theory." He said he was reminded of a walk he one day had with his blind friend. The day was hot and he turned to the blind friend and said, "I wish I had a glass of milk." "Glass," replied the blind friend, "I know what that is. But what do you mean by milk ?" "Why, milk is a white fluid," explained Einstein. "Now fluid, I know what that is," said the blind man. "but what is white ?" "Oh, white is the color of a swan's feathers." "Feathers, now I know what they are, but what is a swan ?" "A swan is a bird with a crooked neck." "Neck, I know what that is, but what do you mean by crooked ?" At this point Einstein said he lost his patience. He seized his blind friend's arm and pulled it straight. "There, now your arm is straight," he said. Then he bent the blind friend's arm at the elbow. "Now it is crooked." "Ah," said the blind friend. "Now I know what milk is."

Thank you for mentioning it. This reminded me about the question I wanted to ask for some time: Is Planck time the shortest time only in combination of GR with QM? In other words, it is the shortest neither in the standard QM nor in the standard GR, but only if GR is extrapolated to very small distances? If so, how certain it is as there is no established theory of quantum gravity? Another question: If tP is a short interval measurable in some rest frame, wouldn't we be able to measure a shorter part of it if we move fast enough relative to that frame, because of the time dilation? (I.e., tP in the rest frame will become γtP in our frame which is >tP)

I don't think so. Let's take Schrödinger equation, for example. It relates time derivative d/dt and spatial derivative d/dx. Thus, it assumes that time interval and distance can be infinitesimally small together.

There is no finite limit to how small a time interval between events can be (in QM, SM, SR, and GR.)

My professors and textbooks in classes on evolution have never mentioned anything like this. What does it mean?