Differential geometry is the study of smooth manifolds—curved spaces that locally look like Euclidean space—and the geometric structures defined on them. Unlike the classical extrinsic approach of studying curves and surfaces embedded in higher-dimensional Euclidean space, differential geometry emphasizes intrinsic properties: those measurements and invariants that depend only on the manifold itself, not on any ambient space. The central realization is that curvature, arising naturally from connections on tangent bundles, provides the language to measure how space curves and is the key to understanding topology through analysis. By developing calculus on manifolds, we learn that differentiation is not an absolute operation but requires a choice of connection—a way of comparing tangent spaces at nearby points.

The course traces a systematic path from foundations to applications. We begin with smooth manifolds and tangent bundles, establishing the local and global coordinate structures that make calculus possible. Differential forms then provide a coordinate-free calculus, leading naturally to curvature as the obstruction to integrability. Connections formalize the notion of parallel transport and covariant differentiation on vector bundles, revealing that curvature is a fundamental invariant living in these connections. Three classical geometric structures—Riemannian metrics, symplectic forms, and complex structures—exemplify how to impose additional constraints on a manifold, each with distinct geometric meaning and topological consequences. Geodesics emerge as the shortest paths on a Riemannian manifold, and their existence and properties reflect the underlying curvature. Finally, the Yang-Mills equation presents a variational principle on gauge theory, showing how connections themselves become dynamical objects satisfying equations of motion, unifying geometry with physics.

Each chapter builds essential machinery for the next: manifolds provide the setting; differential forms give us a calculus; connections supply the notion of differentiation; geometric structures impose regularity; geodesics reveal the interplay of local curvature and global shape; and Yang-Mills demonstrates that these structures obey natural dynamical laws. Together, they form a unified framework in which local infinitesimal invariants determine global geometric and topological properties.

# 1. Smooth Manifolds

Differential geometry begins with the question: what is the right setting for doing calculus on curved spaces? Euclidean space $\mathbb{R}^n$ is the model, but many natural geometric objects — spheres, projective spaces, Lie groups — are not flat. A smooth manifold is the answer: a topological space that locally looks like $\mathbb{R}^n$, equipped with enough structure to make calculus well-defined. This chapter builds that structure from the ground up, introduces the three equivalent constructions of the tangent space, develops the language of vector bundles, and proves the Frobenius integrability theorem, which tells us precisely when a family of tangent directions can be integrated into submanifolds.

## Topological and Smooth Manifolds

The first step is to identify what topological properties are needed to do geometry sensibly.

[definition: Topological Manifold] A **topological manifold** is a Hausdorff, second countable topological space $X$ that is locally homeomorphic to $\mathbb{R}^n$ for some fixed $n$, called the **dimension** of $X$. Second countability means there exists a countable basis $\{U_i\}_{i \in \mathbb{N}}$ for the topology of $X$. Hausdorff means distinct points can be separated by disjoint open sets. [/definition]

The Hausdorff and second countability conditions are not merely technical: without Hausdorff, limits are not unique and the geometry becomes pathological; without second countability, partitions of unity (a key tool) need not exist.

A **chart** centred at $p \in X$ is a pair $(U, \varphi)$ where $U$ is open in $X$, $p \in U$, and

\begin{align*} \varphi : (U, p) \xrightarrow{\;\sim\;} (B^n, 0) \end{align*}

is a homeomorphism onto the open unit ball $B^n = B(0,1) \subset \mathbb{R}^n$. A choice of chart near $p$ gives **local coordinates** $\{x_1, \ldots, x_n\}$ on $X$ near $p$, via the pullback of standard coordinates on $\mathbb{R}^n$. Given two charts $(U, \varphi)$ and $(V, \psi)$ at $p$, the **transition function** is

\begin{align*} \psi \circ \varphi^{-1} : \varphi(U \cap V) \longrightarrow \psi(U \cap V), \end{align*}

which is a homeomorphism between open subsets of $\mathbb{R}^n$. Since the target and domain lie in $\mathbb{R}^n$, it makes sense to ask whether this transition function is smooth. Requiring smoothness is precisely what promotes a topological manifold to a smooth manifold.

[definition: Smooth Manifold] A **smooth (differentiable) manifold** is a topological manifold $X$ equipped with an **atlas** of charts $\{(U_\alpha, \varphi_\alpha)\}_{\alpha \in A}$ — an open cover of $X$ — such that all transition functions \begin{align*} \varphi_{\alpha\beta} = \varphi_\beta \circ \varphi_\alpha^{-1} : \varphi_\alpha(U_\alpha \cap U_\beta) \longrightarrow \varphi_\beta(U_\alpha \cap U_\beta) \end{align*} are smooth diffeomorphisms for all $\alpha, \beta$. A choice of such an atlas is called a **differentiable structure** on $X$. [/definition]

Any smooth atlas is contained in a unique **maximal atlas**, obtained by including every chart whose transition maps with the given atlas are smooth (this maximality is proved via a Zorn's lemma argument). Two atlases that determine the same maximal atlas define the same differentiable structure. Remarkably, the same topological manifold can admit distinct differentiable structures: Milnor's exotic spheres are the classical example.

We write $M^n$ for a smooth $n$-dimensional manifold.

[definition: Smooth Map] If $M, N$ are smooth manifolds, a map $f : M \to N$ is **smooth at** $p \in M$ if for any chart $(U, \varphi)$ at $p$ and chart $(V, \psi)$ at $f(p) \in N$, the composition \begin{align*} \psi \circ f \circ \varphi^{-1} : \varphi(U) \longrightarrow \psi(V) \end{align*} is smooth (in the classical sense, as a map between open subsets of Euclidean spaces). We say $f$ is **smooth** if it is smooth at every $p \in M$. This notion is independent of the choice of charts, precisely because the transition maps are smooth. [/definition]

With smooth maps defined, the correct equivalence relation on smooth manifolds is diffeomorphism: a smooth bijection with smooth inverse. This is the natural isomorphism in the category of smooth manifolds — two diffeomorphic manifolds are geometrically indistinguishable.

[definition: Diffeomorphism] Smooth manifolds $M, N$ are **diffeomorphic** if there exists a smooth bijection $f : M \to N$ with smooth inverse. [/definition]