What exactly is differential geometry?

[pierce15]

Does it still have a sense of Euclid-style geometry-are there still cubes and spheres, so to speak? Is it mostly about 1D curves/2D surfaces, or does it consider higher dimensions? Are the surfaces which the field concerns mostly graphs of several variables, e.g. ## x^3+y^3+z^3=1 ##, or are they more abstract, like in topology? What prerequisites does it have? Are complex numbers/complex analysis used at all?

 

[jedishrfu]

Differential geometry covers a lot of things:

http://en.wikipedia.org/wiki/Differential_geometry

General relativity utilizes it in describing gravitation as a manifestation of curved spacetime.

 

[micromass]

Does it still have a sense of Euclid-style geometry-are there still cubes and spheres, so to speak?

While it's certainly possible for cubes and spheres to show up, they are not central to differential geometry. It certainly has a very different flavor than usual Euclidean geometry.

Is it mostly about 1D curves/2D surfaces, or does it consider higher dimensions?

Higher dimensions are extremely important in differential geometry. However, when you first start out, you will usually only learn about curves and surfaces. Only later do they really consider higher dimensions.

Are the surfaces which the field concerns mostly graphs of several variables, e.g. ## x^3+y^3+z^3=1 ##, or are they more abstract, like in topology?

They are defined very asbtractly, much like in topology. In fact, the things considered in differential geometry are called manifold and they are topological spaces with a certain smooth structure.
Graphs do show up and are very important. In fact, we can prove that every manifold is actually (locally) the same as a graph of a good function.

What prerequisites does it have?

An introductory course would be about curves and surfaces. The prerequisites are calculus and linear algebra. A rigorous analysis course wouldn't hurt either.
More advanced courses require a very good knowledge of topology

Are complex numbers/complex analysis used at all?

Yes, they are important in many ways. In fact, an entire branch of differential geometry is focused on complex geometry.

 

[pierce15]

Thanks a lot for the answers, micromass. While you're here, I have another question: is differential geometry related to algebraic geometry?

 

[micromass]

Thanks a lot for the answers, micromass. While you're here, I have another question: is differential geometry related to algebraic geometry?

There are many connections between the two fields. Many ideas and concepts in algebraic geometry and differential geometry are the same, but the technical details can look very different. There are also nice results which actually transform an algebraic situation in a differential situation.

The difference between the two fields is roughly that they study different objects. For example, differential geometry studies things that are smooth everywhere. A thing that is typically not studied is the subset of ##\mathbb{R}^2## that is the union of the ##x## and the ##y## axis. It is not studied because there is a problem in ##(0,0)## (in all points of the space, you can go in two directions, but at the origin you can go in four directions). However, such singular points are studied in algebraic geometry. So algebraic geometry can deal with things that are not smooth. On the other hand, algebraic geometry typically only caress about polynomials. So you want your spaces to be defined by polynomials.

 

[mathwonk]

differential geometry is the study of a (usually) smooth manifold equipped with a smoothly varying dot product on its tangent spaces. algebraic geometry is the study of zero loci of polynomials. here there is no given metric, and the spaces are more restricted in one sense, by being defined by polynomials rather than more general smooth functions, but less general in another sense in that they need not be smooth.