TITLE: Differential topology versus differential geometry
QUESTION [47 upvotes]: I have just finished my undergraduate studies. During last two semesters I've taken two subjects dealing with manifolds:

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
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.

REPLY [7 votes]: 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".