> Could you elaborate a bit on this?
Please correct me if I am wrong, I have not touched this subject in a long time and only have some intuition. Here is how I understand it:
A manifold is a kind of space that looks flat when you zoom in close enough. The surface of a sphere or a doughnut is a 2D manifold, and the space we live in is a 3D manifold. A Calabi Yau is one of these spaces but with more dimensions and extra symmetry that makes it very special.
In geometry there are several ways to describe curvature. The most complete one is the Riemann curvature tensor, which contains all the information about how space bends. If you take a specific kind of average of that, you get the Ricci curvature tensor. Ricci curvature tells you how the size of small regions in space changes compared to what would happen in flat space.
Imagine a tiny ball floating in this curved space. If the Ricci curvature is positive, nearby paths tend to come together and the ball’s volume becomes smaller than it would in flat space. If the Ricci curvature is negative, nearby paths move apart and the ball’s volume grows larger. If the Ricci curvature is zero, the ball keeps the same volume overall. So when I said “the space does not stretch or shrink overall” I was describing this situation: the Ricci curvature is zero, which means the space does not expand or contract on average compared to flat space.
The space can still have complicated twists and bends. Ricci curvature only measures a certain type of curvature related to volume change. Even if the Ricci tensor is zero, there can still be other kinds of curvature present. The curvature balances out is just an intuitive way to express that the volume effects cancel when you take the average that defines Ricci curvature. It does not mean the space has matching regions of positive and negative curvature in a literal sense, but rather that the mathematical combination producing Ricci curvature sums to zero.
Noe back to definition: A Calabi-Yau manifold is defined as compact (finite in size), complex), and Kähler (it has a compatible geometric and complex structure), with a first Chern class equal to zero. Yau’s theorem proves that such a space always has a way to measure distances so that its Ricci curvature is exactly zero. So when I said “the curvature balances out perfectly so the space does not stretch or shrink overall” I meant it as an intuitive description of this Ricci flat property. The space is not flat like a sheet of paper, but its internal geometry is perfectly balanced in the sense that there is no net expansion or contraction of space.
