Differential topology versus differential geometry

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.

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".