*the evolutionary process. I draw attention to two points that Tao makes in the passage quoted below.*

__read time from__*time vector field*which obeys the transversality condition " gives a more precise generalization of the "directionality" of time, but this is only the beginning of the journey...

*(* Mammals and reptiles do not refer to mathematicians but to the unnamed ;)*

*time coordinate*, as well as a

*time vector field*which obeys the transversality condition . The level sets of the time coordinate t then determine the sets M(t), which (assuming non-degeneracy of t) are smooth d-dimensional manifolds which collectively have a tangent bundle which is a d-dimensional subbundle of the d+1-dimensional tangent bundle of . The metrics g(t) can then be viewed collectively as a section of . The analogue of the time derivative is then the Lie derivative . One can then define other Riemannian structures (e.g. Levi-Civita connections, curvatures, etc.) and differentiate those in a similar manner.

The former approach is of course a special case of the latter, in which for some time interval with the obvious time coordinate and time vector field. The advantage of the latter approach is that it can be extended (with some technicalities) into situations in which the topology changes (though this may cause the time coordinate to become degenerate at some point, thus forcing the time vector field to develop a singularity). This leads to concepts such as *generalised Ricci flow*, which we will not discuss here, though it is an important part of the definition of *Ricci flow with surgery *(see Chapters 3.8 and 14 of Morgan-Tian’s book for details)."