Differential geometry is the language of modern physics as well as an area of mathematical delight. In particular, this includes classical studies of the curvature of curves and surfaces. The remaining three areas are collectively known as Topology. Widely applicable in geometry and analysis, topology also allows for some bizarre examples and set-theoretic conundra. They include familiar surfaces.
Cell complexes are spaces made of pieces which are part of Euclidean space, generalizing polyhedra.
These are the most classic spaces on which groups of transformations act. This is also the setting for knot theory. You might want to continue the tour with a trip through analysis .