MAT5600 Differential Geometry Spring 2008 subtopic: Einstein for Dummies? Instructor: Robert Jantzen

Course Catalog Description:

What is this stuff?

Differential geometry is the mathematics used to describe curved spaces, building on the foundations of multivariable calculus.

For the very first time, this course will be an elementary introduction to differential geometry that will not only to deal with the geometry of curves and surfaces but will have an eye towards simple 2-dimensional applications to the flat geometry of special relativity and to the curved geometry associated with a spherical black hole. Problem sets will be assigned throughout the semester to make lectures, notes and readings more concrete through calculation, with some guided assistance from Maple when helpful.

I am thinking that this course will start off by revisiting how one treats the geometry of curves and surfaces in Multivariable/Vector Calculus, and by slightly adjusting the familiar details (we'll remember them!), one can extrapolate easily to handle both curved spaces and see how the flat hyperbola geometry associated with the hyperbolic functions usually ignored in calculus describes very nicely the heart of special relativity (time dilation, length contraction, Lorentz transformations). Similarly building on the use of polar, cylindrical and spherical coordinates in flat space, one can easily handle curved spaces with the same tools, enabling one to understand the usual suggestive surface representations of the deformation of the Euclidean plane by the presence of a black hole [see the illustration above above]. You don't have to be an Einstein to comprehend these elementary ideas! We will try to have some fun.

Some history [more depth]

Newton and Liebnitz were the leading thinkers behind calculus back in the 17th century (the 1600s!). Although the curvature of a curve was contemplated long before calculus entered the picture, and afterwards as well, it took some time before calculus was used to explore the complete geometry of curves by a couple of French mathematicians Frenet and Serret around 1850 and it was even later that this was presented in the vector calculus style of modern textbooks. This geometry is important in elementary physics to describe centripetal accelerations and centrifugal forces that we all (well most of us) have direct experience with from driving cars around curves in the road.

However, once curves were investigated, it was natural to move on next to surfaces. In the mid 1800s Gauss, one of the most prolific mathematicians of all time, pioneered this field, but it was his student Riemann who got serious about curved spaces. Unfortunately, he did not live long enough to fully develop the tools of Riemannian geometry, which was left to others, most notably some Italian mathematicians Ricci and Levi-Civita who were the principle thinkers behind tensor calculus at the turn of the 19th (to 20th) century, ready just in time for Einstein to apply their machinery to his curved spacetime theory of gravitation called general relativity. Einstein, however, as a physicist who did not like to write lots of equations, was not big on abstract mathematics and had to be introduced to this stuff by a friend who was a mathematician (Grossmann) in the early 1900s. After some rumination time, general relativity was born in 1916. This pushed curved geometry into the public imagination and physicists continued using this kind of geometry to unify their force fields for the rest of the 20th century. String theory is the latest part of the program to geometrize the forces of nature.

Interestigly enough GPS (gravitational positioning system) is an acronym which has quickly become a part of our daily life, most commonly in car navigational systems, but more and more available in our cell phones and other technology devices. GPS would not work without special and general relativistic effects that differential geometry provides the calculational tools for evaluating.

However, you don't have to be interested in describing the fundamental forces of nature to use differential geometry—it has lots of more mundane applications. For example a flower that is shaped like a trumpet horn initially growing outward from the base is symmetric about an axis of symmetry, but as it bends outward while growing, eventually its curvature will cause it to no longer be rotationally symmetric but develop ruffles.