  A pedagogical but concise overview of Riemannian geometry is provided, in the
context of usage in physics. The emphasis is on defining and visualizing
concepts and relationships between them, as well as listing common confusions,
alternative notations and jargon, and relevant facts and theorems. Special
attention is given to detailed figures and geometric viewpoints, some of which
would seem to be novel to the literature. Topics are avoided which are well
covered in textbooks, such as historical motivations, proofs and derivations,
and tools for practical calculations. As much material as possible is developed
for manifolds with connection (omitting a metric) to make clear which aspects
can be readily generalized to gauge theories. The presentation in most cases
does not assume a coordinate frame or zero torsion, and the coordinate-free,
tensor, and Cartan formalisms are developed in parallel.
