 We see that at any point in n-dimensional space there are n independent directions and each line through a point can have a different curvature. Here we picture just two. In one direction we have a positive curvature, in another it is negative. We can also construct a surface for each dimension. Each surface has a size and an orientation. We can use vectors to mathematically represent the size and orientation of the surfaces. And for each surface we can construct a vector that represents the curvature of the lines through a point on the surface. Multiplying these two vectors creates a mathematical object called a tensor. The power of tensors lies in two basic characteristics. First they carry a great deal of information. And second they are invariant when the coordinate systems are changed. In other words they remain constant across all kinds of changes in how we are looking at any particular situation. Riemann developed a mathematics for this generalized space with any number of dimensions. Basically he came up with a three step process. First we define a metric for the space that allows us to measure distances. Second we use the distance metric to find the geodesics for the space. And third we use the geodesics to define what we mean by curvature. These spaces are smooth. By that I mean there are no abrupt changes. Given that we can always zoom into a curved space to the point that the small piece we are looking at is flat. And in flat space distance between any two points is defined by the Pythagorean theorem. We then generalize by adding coefficients to take into consideration the different scales or lines in different directions. And the fact that the lines no longer cross to form right angles. This is called a metric tensor. And finally we extend the number of dimensions and generalize the coefficients to be functions of a location to take into account curved and changing coordinate systems. This generalized metric tensor is the foundation for non-Euclidean geometry. It's geodesics and it's curvature at any point. With the metric tensor we can measure distances between any two points by adding up all the small distances along the way. Taking the formula and finding its minimum is an exercise in calculus that gives us the shortest distance. These are the geodesic lines. Now that we have a way to measure distance and find geodesics we can determine a space's curvature. Riemann used a concept called parallel vector transport. Picture moving a vector around a triangle in flat space in such a way that it remains parallel to the starting vector. By the time we get back to the start we have the exact same vector as we started with. Now repeat this same exercise on a curved surface like the surface of the earth. Start at the equator and point the vector in front of you facing north. Move north along the geodesic longitudinal line. When you reach the north pole turn 90 degrees to the right. To keep the vector pointing in the same direction it is now pointing to your left 90 degrees. Move south along the geodesic longitudinal line to the equator and turn right 90 degrees again. The vector must now be made to point to your rear in order to keep it parallel to the original direction. Now walk west along the equatorial geodesic until you reach your starting point. Clearly the vector is no longer the same as when you started. The difference is the measure of the curvature of the space you traveled. Riemann developed the tensor that precisely measures how much the components of a vector change when it is parallel transported along a small closed curve. This is called the Riemannian curvature tensor. A subset of this tensor was developed by a mathematician named Gregorio Ricci-Curbostro. Where Riemann gives us the curvature for every geodesic. Ricci gives us the average for a volume. With this we can calculate the amount by which a volume deviates from what it would be in Euclidean space. For example, in Euclidean flat space a cuboid's volume is A times B times C. The yellow lines represent geodesics inside the box. The volume is less than this if the Ricci curvature in the interior region is positive. In other words it's smaller on the inside. The volume is more than this if the Ricci curvature of the interior region is negative. In other words it's larger on the inside.