Flatness means that two parallel lines on a flat 2d object never meet and never get farther apart from each other. Two lines can also never meet on a hyperbolic surface (think of a saddle for horseback riding) but after a single point on each line that is as close as the lines ever get, they always get farther apart after that. Two parallel lines on a sphere, while seeming to be parallel after they travel an infinitesimal distance, always end up meeting at the poles. All of the lines with intersect the poles are also known as great circles. Lines of longitude are all great circles, as is the equator. There can be an infinite number of pole pairs and an infinite number of sets of great circles on any given sphere.
A flat 2d surface that is infinite in one direction and finite in the direction perpendicular to that can be rolled up into a cylinder in which the two lines travel around the short direction to make two circles, or travel around the cylinder in two parallel spirals forever and ever, or travel parallel to each other in a direction parallel to the infinite direction.
A cone is another ‘flat’ shape, or more properly, a shape which can be made by cutting up a flat sheet and taping it together. Think of a disk with a section missing. If you take a paper cone such as one suitable for a sno-cone and cut from the edge down to the point, you will end up with a disk that has a missing section. Parallel lines will behave similarly to the infinite spirals on a cylinder except that after coming close to the point they will cross both their parallel partner and themselves at points. The behavior of the parallel lines will be similar to the behavior of a ribbon or a piece of tape attached to a cylinder or a cone.
When one is dealing with two infinitesimally close parallel lines, lines that are traveling as close to straight as they can be made to be, one is said to be traveling along a geodesic.
A flat 2d metric can always be reduced to a square 2 by 2 ‘indicator’ (I forget the exact term) matrix such as the following:
(1, 0)
(0, 1)
A flat universe with one time dimension and 3 space dimensions can always be reduced to either a 1 with 3 negative ones or a -1 with 3 positive ones along the trace of the ‘indicator’ matrix, such as the following:
(1, 0, 0, 0)
(0, -1, 0, 0)
(0, 0, -1, 0)
(0, 0, 0, -1)
or
(-1, 0, 0, 0)
(0, 1, 0, 0)
(0, 0, 1, 0)
(0, 0, 0, 1)
Here is a flat space-time metric in Cartesian coordinates which would approximate an empty vacuum in our universe which was far away from any mass:
c^2 dt^2 - dx^2 - dy^2- dz^2
where c is the speed of light, dt is the time infinitesimal, and dx, dy and dz are the infinitesimals for the x, y, and z directions respectively.
Whether you ultimately end up with a 1 and three -1s or a -1 and three 1s along the trace depends upon which convention you favor for the metric. Once you pick a convention, you have to stick with it throughout your calculations. After that, you cannot change conventions (or horses) in midstream.
In a flat space-time such as ours, the product of the trace (in this case, equal to the determinant) of the ‘indicator’ matrix must always turn out to be -1.