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Generally, events are analyzed by the use of three spatial coordinates and one time coordinate. In this discussion, events are only analyzed by the use of one time coordinate.

This method can simplify the analysis tremendously. The time coordinate is assumed to take a form of geometry similar to physical space. This is the extraction of time from the spacetime of relativity and keeping the spatial parts hidden away. In this way, one can define time as a line embedded in the following three geometries: Euclidean (planar), Riemannian (spherical) and Lobatchevskian (hyperbolic).

By parallel timed events, it is to mean that simultaneity can be defined between one event in one timeline and another event in the other timeline. But for an observer in one timeline, he or she is not capable of knowing whether these two events are simultaneous. There is no information send by photons from one to the other. These events can only be assumed to be simultaneous. And that is all one can do. Each photon of each timeline travels along each line and no reason to jump to the other line. Each line contains its own reality.

By serial events, it is to mean that simultaneity has no meaning. All events follow or lead other events. These successions of events gave true meaning to the word “timeline.”

In Euclidean geometry, two parallel timelines can be constructed. The simultaneity of two parallel events is the end points of the perpendicular distance between the lines. In theory, there are an infinite number of simultaneous events. But the following assumptions must be made that (1) the time tick-marks are the same for the lines and (2) that the line are infinitely extended in both directions. The events along one timeline are all serial events.

In Riemannian geometry, parallel events cannot be constructed. Hence simultaneity has no meaning. All events are serial. But a zero time can be defined as located at one of the poles. The other pole will be the end of time.

In Lobatchevskian geometry, many infinitely extended parallel timelines can be constructed. Each of these timeline contains serial events. But there is only one occurrence of simultaneity for these parallel events. And this happens at the same time.