A curved 4-surface is nearly impossible to imagine for those of us used to believing our ordinary 3-dimensions of space are perfectly Euclidian (or flat).  We have no problems, however, imagining a curved lesser dimensional space imbedded within.  For example, a ball is the perfect example of a curved 2-surface embedded within the flat 3-space we know defined by length, width and height.  In 1851, Georg Riemann posed a doctoral thesis that could mathematically describe curved surfaces and spaces such as these.  He unfortunately never thought of space-time actually being curved, only space.  Einstein studied this math and utilized it to describe General Relativity in 1915.  This is tensor math, and is employed in many papers involving relativity.  Tensor math gets pretty complicated, even for math wizards, but the basic underlying structure of it is pretty fundamental.  A tensor is typically a linear function designed to take a certain combination of input data, and convert this data to a specific set of output data.  The perfect example of a tensor is the Riemann curvature tensor.  This tensor takes three inputs, the 4-velocity of test particle along a certain geodesic, a small separation to a neighboring geodesic, and the 4-velocity of the same test particle along this neighboring geodesic.  First, lets take a step back; the reader should be familiar with velocity from Distortions of Relativistic Measure (Schoedl, 2003).  A 4-velocity is just an ordinary 3-vector (direction in 3-space), times the speed.  This is termed a 4-velocity, because the direction and speed of the particle is defined as a combination of the 3 dimensions of space, and one of time.  Geodesics you learned in 4^th grade Geography, hopefully.  Hold a string between two points on a globe, and pull the string tight.  This forms the shortest distance between two points on the globe, and geodesics are the shortest distance between two points in space-time.  In the case of the globe of Earth, most of these geodesics appear curved when you draw them onto flat maps.

Consider a thought experiment, Changes in Latitude.  Since the globe is a 2-surface, a speed along one of its geodesics could technically be considered a 3-velocity.  Imagine the clock Tik zipping from the North Pole to the South Pole along the 90º line of longitude through Chicago.  Let the clocks be 2-dimensional and confined to the surface of the globe.  Now we need a second clock Toc zipping along a neighboring line of longitude near the first clock, say 75º through New York.  The clocks will start out moving at an angle slightly away from each other.  They will end up parallel at the