# Differential topology versus differential geometry

I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds:

1. Analysis on manifolds, containing: definition of manifold, tangent space (as derivations and classes of curves), vector fields, vector bundles, flows, Lie derivatives, integration on manifolds (Stokes theorem), forms, Hodge decomposition theorem

2. Introduction to differential geometry (for me it should be called introduction to Riemannian manifolds) containing: tensor calculus introduction, Riemannian manifold definition, connections, curvatures, geodesic, normal coordinates, geodesic completness theorem, classification throught curvature, jacobi fields, harmonic maps.

Now, for me differential geometry was/is a theory about manifolds, so anything dealing with manifolds is branch of differential geometry. On the other hand I am preparing for taking part in local conference called: "algebraic and differential topology". I have read: I- books references, II- books references, III- wikipedia, yet I still don't really know if differential topology is subtheory of differential geometry or is it seperate theory and how is it located between differential geometry and algebraic topology. For example I expect that studies on Riemmanian manifolds are part of differential geometry but would problem of classification manifolds up to diffeomorphism be a part of differential topology or geometry ?

Request: I would be grateful for your characterisation of differential topology and differential geometry possibly with examples of problems, theorems present at them.

• Differential topology deals with the study of differential manifolds without using tools related with a metric: curvature, affine connections, etc. Differential geometry is the study of this geometric objects in a manifold. The thing is that in order to study differential geometry you need to know the basics of differential topology. I don't know exactly where the line between them is drawn, but they clearly overlap without one being a subtheory of the other. Jul 5, 2015 at 22:39
• A counterexample to "anything dealing with manifolds is a branch of differential geometry" is that there are topological manifolds that cannot be given the structure of a differentiable manifold, so differential geometry doesn't really apply to those manifolds. Jul 5, 2015 at 22:58
• Try reading the introduction of this book, which I used as a graduate student: amazon.com/Differential-Topology-Graduate-Texts-Mathematics/dp/… (The introduction pages are available in the preview.) Jul 5, 2015 at 23:14
• look at a preliminary ideas at en.wikipedia.org/wiki/… Jul 5, 2015 at 23:58
• @janmarqz actually I've cited it in my post. Jul 6, 2015 at 9:19

First of all, the concept of a "manifold" is certainly not exclusive to differential geometry. Manifolds are one of the basic objects of study in topology, and are also used extensively in complex analysis (in the form of Riemann surfaces and complex manifolds) and algebraic geometry (in the form of varieties).

Within topology, manifolds can be studied purely as topological spaces, but it is also common to consider manifolds with either a piecewise-linear or differentiable structure. The topological study of piecewise-linear manifolds is sometimes called piecewise-linear topology, and the topological study of differentiable manifolds is sometimes called differential topology.

I'm not sure I would necessarily describe these as distinct subfields of topology -- they are more like points of view towards geometric topology, and for the most part one can study the same geometric questions from each of the three main points of view. However, there are questions that only make sense from one of these points of view, e.g. the classification of exotic spheres, and there are certainly topology researchers who specialize in either piecewise-linear or differentiable methods. Differential topology can be found in position 57Rxx on the 2010 Math Subject Classification.

Differential geometry, on the other hand, is a major field of mathematics with many subfields. It is concerned primarily with additional structures that one can put on a smooth manifold, and the properties of such structures, as well as notions such as curvature, metric properties, and differential equations on manifolds. It corresponds to the heading 53-XX on the MSC 2010, and the MSC divides differential geometry into four large subfields:

• Classical differential geometry, i.e. the study of the geometry of curves and surfaces in $\mathbb{R}^2$ and $\mathbb{R}^3$, and more generally submanifolds of $\mathbb{R}^n$.

• Local differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a local point of view.

• Global differential geometry, which studies Riemannian manifolds (and manifolds with similar structures) from a global point of view.

• Symplectic and contact geometry, which studies manifolds that have certain rich structures that are significantly different from a Riemannian structure.

As a general rule, anything that requires a Riemannian metric is part of differential geometry, while anything that can be done with just a differentiable structure is part of differential topology. For example, the classification of smooth manifolds up to diffeomorphism is part of differential topology, while anything that involves curvature would be part of differential geometry.

• I don't know if the MSC does this, but it'd probably also be fair to add a section for complex differential geometry, which tends to be both similar and differently flavored than both symplectic and Riemannian geometry. (Also, I'm not even sure your last sentence is completely fair! Is studying what manifolds support metrics of positive scalar curvature - which turns out to be equivalent in certain dimensions to the vanishing of a (modified) $\hat{A}$-genus - geometry, and not topology?)
– user98602
Jul 6, 2015 at 0:09
• @MikeMiller The MSC has a few sub-headings under both local and global differential geometry regarding complex manifolds as well as a separate "Analytic Spaces" sub-heading under the separate "Several Complex Variables and Analytic Spaces". Regarding your latter comment, I would say that my last sentence is broadly true, though there are certainly exceptions. It's hard to be completely fair when making these sorts of sweeping statements! Jul 6, 2015 at 2:44
• Sure! I broadly agree with your answer - I just wanted to mention that there's subtlety all the way down, so to speak.
– user98602
Jul 6, 2015 at 3:17
• I find your answer very helpful and I am grateful for it! Jul 6, 2015 at 9:28

The basic objects in differential geometry are manifolds endowed with a metric, which is essentially a way of measuring length of vectors. A metric gives rise to notions of distance, angle, area, volume, curvature, straightness and geodesics. It is the presence of a metric that distinguishes geometry from topology. However, another concept that might contest for the primacy of a metric in differential geometry is that of a connection. A connection in a vector bundle may be thought of as a way of differentiating sections of the vector bundle. A metric determines a unique connection called the Riemannian connection with certain desirable properties. While a connection is not as intuitive as a metric, it already gives rise to curvature and geodesics. With this, the connection can also lay claim to be. a fundamental notion of differential geometry.

This is from preface of Loring W. Tu's book titled "Differential geometry - Connections, curvature and characteristic classes".