As a pc scientist studying maths I’d endlessly be selecting up new and thrilling books solely to be misplaced on the primary web page. Over time I ultimately picked up sufficient ideas to get by way of, nevertheless an off-the-cuff, visible instinct would have made the journey far simpler. That’s what this weblog sequence goals to attain.
For this journey I might be loosely following this book [1] because it offers an intensive overview of the topic, steadily cited because the “go to” for Riemannian Geometry. The one prerequisite is the fundamentals of set principle and calculus.
Let’s go over a couple of core ideas that might be important for mastering Riemannian Geometry. The primary time we come throughout a time period from this glossary it’ll be in daring.
Notation
x — A mapping
p — A degree
f :X ↦ Y— A perform
U — A subset
V — A Subset
Injective/bijective:
An injective perform maps distinct components of it’s area to distinct components. Basically, a one-to-one mapping the place totally different inputs get totally different outputs.A perform is Bijective if every aspect of the area maps onto precisely one aspect of the codomain.
Neighborhood:
A set of factors round and containing a given level, the place one can transfer round in any path away from the purpose with out leaving the set.
Homeomorphism:
Comes from the greek “comparable form”, and is a particular sort of mapping which preserves topological properties.
Diffeomorphism:
A perform is a diffeomorphism (assume differentiable-morphism) whether it is differentiable map and it’s inverse can also be differentiable.
Parameterization:
The inputs a perform/object is dependant on. As an example t is a parameter of f(t).
