"Curved geometries are in the domain of
Non-Euclidean geometry. An example of a positively curved space would be the surface of a sphere such as the Earth. A triangle drawn from the equator to a pole will have at least two angles equal 90°, which makes the sum of the 3 angles greater than 180°. An example of a negatively curved surface would be the shape of a
saddle or mountain pass. A triangle drawn on a saddle surface will have the sum of the angles adding up to less than 180°."
en.wikipedia.org
There are several different things to unpack here.
First, an n-dimensional manifold is one that looks like n-dimensional Euclidean space 'up close'.
So, the surface of the Earth looks flat' close up, in other words like a 2-dimensional plane. So the surface of a sphere is a *2* dimensional manifold. Another way to see this is that it takes two coordinates (latitude and longitude) to locate a point on the surface of the Earth.
Space as we usually think about it 'looks like 3 dimensional Euclidean space' up close, so it is a *3* dimensional manifold. So, it takes 3 coordinates to locate a point in space (1: how far up and down, 2: how far left or right, 3: how far front or back).
Spacetime requires 4 coordinates (3 for space and one for time) to locate a point, so is a 4 dimensional manifold.
Next, if we look at latitude lines on the surface of a sphere, we get circles, which are a 1 dimensional manifold. These 'cross sections' where you fix one coordinate are submanifolds that have a dimension one less that the overall manifold.
So, space at a fixed time is a 3 dimensional submanifold of the 4 dimensional manifold of spacetime.
As another example, if we look at usual 3 dimensional Euclidean space, but use the radius from some point as one of the coordinates (and maybe latitude and longitude for the others), we find the submanifolds to be spheres of 2 dimensions. Furthermore, larger radii correspond to larger spheres.
For an expanding universe, we looks at different cross sections for spacetime at different times. All that it means to be expanding is that the distances in those cross sections are larger at later times.
So, we can look for the 'shape of space' as well as the 'shape of spacetime'. General relativity relates the curvature of spacetime to the density of mass and energy. So, more mass produces more curvature in spacetime.
The pictures you showed are 2 dimensional *analogies* for 3 dimensional space as cross sections of 4 dimensional spacetime. The balloon analogy is also a 2 dimensional analogy for 3 dimensional space.
Next, a sphere is said to be 'positively curved' and a saddle shape is 'negatively curved'. These describe 2 dimensional manifolds, but there are analogous concepts for higher dimensional manifolds. Under general relativity and in a uniform universe, the curvature of space changes over time as the universe expands. This curvature is related directly to the expansion factor.
Extensive measurements have been made over the last century or so to determine whether space is positively curved, negatively curved, of not curved at all. What we have found is that the curvature is too small to detect. Since a very small curvature cannot be distinguished from zero curvature, we don't know which of the three possibilities actually holds. This is one of the great open questions in cosmology. It is important because a positively curved universe is expected to contract eventually, potentially leading to a 'bounce' that can repeat. Zero and negative curvatures don't have that possibility (in the same way).
Mathematicians have classified the 2 dimensional manifolds. We are actively working on 3 dimensional manifolds (the Poincare conjecture is related to this) and we know that there is no way to completely classify manifolds of dimension 4 or higher.
