About
Differential geometry can be understood as what happens if geometry
is investigated using tools from differential calculus.
The most common interpretation is the study of Riemannian manifolds (M,g)
that is manifolds on which a metric structure is defined using a metric
tensor g. This is a fancy way to say that we have at each point of the
manifold an inner product. Unlike in multi-variable calculus, which is
the case when g is the identity matrix and M is either the two dimensional
plane or three dimensional space, the dot product and so the measurements
can depend on the point. This is one of the key ideas in general relativity:
mass defines the geometry of space and mass points move on geodesics in that
geometry.
