Linear Algebra - A basic knowledge of linear algebra is essential to the study of differential geometry. The linearisation of a smooth map features heavily throughout differential geometry as do determinants, traces, eigenvalues and eigenvectors. Many of the fundamental objects studied in differential geometry are tensor fields which incorporate and extend undergraduate linear algebra.

Topology - A basic knowledge of topology is very useful in the study of differential geometry - it provides a language and structure for notions relating to continuity and allows us to generalise from Euclidean submanifolds to manifolds as objects in their own right. Our focus is in what is often called point set topology. We won't be so interested in algebraic topology, and in particular we won't spend much time discussing coffee cups and donuts.

Differential Calculus - Differential calculus is essential to the study of differential geometry - that's the differential part!

Integral Calculus - Integral calculus is essential to the study of differential geometry - it's the counterpoint to differential calculus and is as important in differential geometry as it is in calculus.

Euclidean Sub-manifolds - Certain subsets of Euclidean space admit a suitable theory of calculus. The class of subsets we consider are the so-called sub-manifolds. Linear subspaces are a particular example, but not all (in fact most) sub-manifolds are not linear subspaces.

Inverse Function Theorem - The Inverse Function Theorem is a cornerstone theorem in nonlinear analysis. It is essential for the development of sub-manifolds. There are several equivalent formulations such as the Implicit Function Theorem, the Immersion Theorem and the Submersion theorem. We will discuss each of these and see that they are in fact equivalent to each other. We won't focus too much on the proof itself however as this is result best dealt with in an analysis course.

Embedding of Sub-manifolds - Let us now investigate the manner in which an embedded sub-manifold is embedded in Euclidean space. There is a compatibility between the intrinsic differentiable structure from local parametrisations and the ambient differential structure from the surrounding Euclidean space. We will also see how the regularity condition of injective differential is precisely what is required to allow us to develop a theory of calculus on an embedded sub-manifold. The main tool used is the *Inverse Function Theorem* and several variants such as the *Implicit Function Theorem*.

Manifolds - Historically, it was some time before mathematicians realised the importance of the transition maps in defining manifolds. The origins of manifolds date back to Gauss and Riemann in the 1800's. It wasn't until 1913 that Weyl gave a modern definition of surfaces, and then not until 1936 that Whitney gave the current definition of manifolds in arbitrary dimensions.

Smooth Functions - Now we define smooth functions between manifolds. This obviously an necessary definition in order to do any calculus. The approach is to define differentiability in terms of differentiability via charts, which reduces the problem to smoothness in Euclidean space which we already understand. We will also introduce tangent vectors and differentials of smooth maps.

Tangent Vectors - For embedded sub-manifolds in Euclidean space, a tangent vector is the velocity vector (as a vector in the ambient Euclidean space) of a smooth curve lying on the sub-manifold. In order to make this definition, we need to use the ambient Euclidean structure. For a general manifold such a structure is not available, hence we need a way to generalise tangent vectors. The approach taken here is to observe that many curves may represent the same tangent vector, and our definition is to essentially define a tangent vector to be all curves with the same velocity vector in a chart! Next we can define the differential of a smooth function by mapping curves to curves by the function. In a chart, this notion recovers the usual notion of directional derivative.

Differentials of a Smooth Map - With a suitable definition of tangent vector, we can now define directional derivatives and the differential of a smooth function. By mapping curves to curves we define the differential applied to tangent vector as the equivalence class of curves mapped to by the function. When checking this is well defined independently of the choice of curve representing a tangent vector, we recover the Euclidean differential with respect to a chart, giving further justification to the definition of tangent vector.

Tangent Bundle - The tangent bundle is the set of all tangent vectors on a manifold. Contrast this with the tangent space at a point, which is just the tangent vectors based at that point. Thus the tangent bundle is the union of all tangent spaces. The tangent bundle inherits a manifold structure and this allows us to define the notion of vector field.