ChapterZero

Today Dr. Bao gave the math department’s weekly ‘Graduate Student’ Seminar, which it seems like graduate students never give. He spoke about the relation between geodesics and metrics– a different perspective on constructing metrics, the problem of determining the geodesics of a given metric, and the inverse problem of determining a metric given the geodesics.

In relation to the last problem, he showed that the problem of geodesics to metrics is naturally suited to Banach spaces rather than Hilbert spaces, since not all geodesics give rise to a true norm, much less one that is induced by an inner product.

He used only three examples to illustrate his point. The first, he used to show that it can be fruitful and interesting to consider a metric defined in time instead of speed. Specifically, if is a strictly positive function on the domain being considered and is the standard inner product in your space, then
is a norm with properties that depend upon where the vector is anchored. The first example was
, where is the standard norm, and the space being considered is the open ball around the origin of radius 2.
In this universe, it takes a really long time to visit your neighbors if you and they live on the outside of the ball. The geodesics in this space are radial straight lines and arcs on the surface of balls centered outside of the universe that intersect it at right angles. On the other hand, in his second example,
,
the farther from the origin you and your neighbor live, the quicker you can reach each other. I’m going to try to reason out the geodesics in this space.

The final example, on ‘Zermelo Navigation’, disagreed with my intuition, so I’m still puzzling it out. Basically, it is concerned with the fastest paths taken by a person at the center of a rotating disc attempting to reach a fixed point on the perimeter, and attempting to reach the center of the disc from a fixed point on the perimeter. The geodesics in this example are spirals, both ways. (What are the geodesics between points in the interior of the disc? :)) What I don’t get is why, as he stated, it takes less time to go out than to come back in. Also, this is a case of the norm not being induced by any inner product.