Explain branches of geometry for non-mathematician

Question

Some background -
I'm an advanced physics undergrad and lately was motivated to self study basic contemporary geometry to get a better grip on general relativity (maybe there is a more appropriate category for what I call here geometry). I've found many recommendations for books and resources, yet, as an "outsider" to the field of mathematics, I understand very little the difference between the branches of geometry. Wikipedia's definitions didn't help much to get a clear distinction between the different fields, and I'd like to know some more before I invest significant amount of time in this.

The question is - can you describe (or refer to simple description) for a mathematical layman like me what kind of problems the following branches deal with - topology, differential geometry , Riemannian geometry and the differences between them (especially between the latter two).

Answer 1

Topology is really distinct from the other two. Roughly speaking, topology is the study of those properties of a geometric space that do not depend on notions such as distance or angles. Within topology, the surface of a sphere is considered to be the same as the surface of a cube, though these can be distinguished from the surface of a doughnut (i.e. a torus). Topology is also concerned with a much broader class of geometric spaces than the other two, including manifolds, complexes, spaces of functions, and more general topological spaces. The basics of topology are a foundational subject for large portions of mathematics, and most advanced (i.e. graduate-level) treatments of differential geometry assume at least some topology as a prerequisite.

Differential and Riemannian geometry, by contrast, are concerned mainly with manifolds (i.e. curves, surfaces, and their higher-dimensional analogues), and involve notions such as distance, angle, and curvature, as well as the heavy use of differential calculus.

The distinction between differential geometry and Riemannian geometry is subtle.
Riemannian geometry describes a certain approach or point of view towards differential geometry, where the primary objects of study are abstract differentiable manifolds endowed with Riemannian metrics. This approach has been extraordinarily fruitful, and at this point the vast majority of differential geometry clearly falls under the Riemannian framework. Indeed, "differential geometry" and "Riemannian geometry" are sometimes described as synonyms.

At the same time, there are a few subjects within differential geometry that don't really fall within Riemannian geometry. For example, the relatively new fields of symplectic geometry (which is related to Hamiltonian mechanics) and contact geometry study geometric structures that can be placed on a differentiable manifold that are quite distinct from a Riemannian metric, though some would argue that these subjects aren't part of differential geometry either. But even Lorentzian manifolds (quite important in general relativity) aren't strictly speaking Riemannian manifolds, though they are considered "pseudo-Reimannian" and the study of them is quite similar to Riemannian geometry. Finally, folks who study arbitrary connections or even synthetic differential geometry have arguably moved on from Riemannian geometry to something more general.

There are also classical aspects of the study of differential geometry that haven't quite been absorbed into the Riemannian framework, such as the study of curves and surfaces in two and three dimensions. Roughly speaking, Riemannian geometry is concerned with the intrinsic properties of a manifold, so any geometric properties that are extrinsic, i.e. depend on an embedding into an ambient space, don't really fall under Riemannian geometry. Thus Gaussian curvature is a Riemannian concept, but mean curvature and the study of minimal surfaces (i.e. the shapes made by soap films) is really distinct from Riemannian geometry.

Answer 2

Differential Topology and Differential Geometry intersect in a lot of ways from what I've seen. However, if you are more physics literate, you probably speak more fluently in terms of linear algebra and vector analysis which is good for Differential Geometry. Thus, you should probably start there. However, I still find it important to know Topology as it can end up being a very useful subject. You don't really get a good bird's-eye-view of what Topology attempts to study at first, but it eventually starts coming together the more you're exposed. The most important type of topological spaces for General Relativity are finite dimensional manifolds. A good book on this is titled "Differentiable Manifolds and Riemannian Geometry" by Boothby. I know nothing about the difference between Riemannian Geometry and these other two subjects, but my guess would be that Riemannian Geometry has more to do with Complex Analysis, though that's just a guess.

Answer 3

Topology: As a general mind setup, the study of a particular branch of Geometry can be seen as the study of properties of geometrical objects that remain unchanged after a beforehand well-precised set of transformations. For instance, Euclidean Geomety is what you get if you allow only orthogonal transformations of the space: distances are preserved, angles are preserved, lines remain lines and so on. If you allow all affine transformations (an immediate generalizations of linear transformations) instead of just the orthogonal ones, metric notions such as distance or angles lose meaning, but lines remain lines- Topology is what you get when you allow the large set of all bicontinuous invertible transformations of the space. Now also linearity is lost: a line can be transformed into a non straight curve and so on. Yet some basic core properties are preserved: connectedness, compactness and one realizes that some basic properties of continuous functions (e.g. Bolzano's theorem) are toplogical in nature.

Differential Geometry: Object of study of DG are the real and complex varieties (read: geometrical "shapes") that can be described as zero-loci of analytic functions. Here Geometry and Analysis come together, so to speak, allowing the use of analytical tools to study geometrical properties.

Riemannian Geometry: RG is sort of subset of DG: in RG varieties are endowed with an intrinsic notion of distance and curvature. Namely there's little if any reference to an ambient space where these varieties may be embedded. As a (big) example consider Einstein's spacetime of GRT which is locally related to Minkowski's spacetime.

For both DG and RG, the notion of variety (aka, manifold) is fundamental: I suggest to start with a good textbook.