Manifolds: The Language of Modern Geometry

The surface above is a smooth closed 2-manifold rendered inside $\mathbb{R}^3$ for visualisation, deforming through a superposition of spherical harmonic oscillations. The coloured trails are independent Riemannian Brownian motions (geodesic random walks whose generator converges to $\tfrac{1}{2}\Delta_M$ as the step size vanishes), riding the surface in real time; as the metric changes with each deformation, so does the diffusion. The embedding is not part of the manifold's definition: $M$ carries its own smooth coordinates, distance function, and curvature intrinsically, without reference to any surrounding space. A geometer confined to the surface can measure distances, angles, and curvature using only quantities defined on $M$ itself. Rendering it inside $\mathbb{R}^3$ introduces a second, extrinsic notion of curvature (how the surface bends within the ambient space), encoded by the second fundamental form and independent of the intrinsic geometry. Gauss's Theorema Egregium asserts that the Gaussian curvature $K$, though discovered extrinsically, is in fact a purely intrinsic invariant determined by the metric alone.

This is what a manifold is: a space that looks locally flat—every small patch is, in this case, nearly a piece of $\mathbb{R}^2$—but has globally interesting curved geometry. The sphere, the torus, and the space of positive-definite matrices are all manifolds. So is the configuration space of a rigid body and the phase space of a Hamiltonian system. The language of manifolds is the language in which modern geometry, physics, and analysis are written.

The exposition proceeds from topological spaces through smooth structures, tangent spaces, Riemannian metrics, and curvature tensors, ending with the two geometric flows whose mathematics the surface above hints at: curve shortening flow and Ricci flow.

1. Topological Manifolds

1.1. Charts and coordinate maps

The defining property of a manifold is local Euclidean structure: every point has a neighbourhood that looks like an open subset of $\mathbb{R}^n$.

1.2. Why Hausdorff and second countable

Both separation conditions are worth examining. The Hausdorff condition rules out pathological examples like the line with a doubled origin: take two copies of $\mathbb{R}$ and identify all points except the origin. The result is locally Euclidean of dimension 1 but has two distinct "origins" that cannot be separated by open sets.

Second countability ensures that $M$ is paracompact, a technical condition equivalent to the existence of locally finite refinements of any open cover. Paracompactness is used in the partition-of-unity argument that underlies the existence of Riemannian metrics ([riemannian-existence]). Without second countability, the long line is locally Euclidean of dimension 1 but admits no countable atlas and no partition of unity.

1.3. Examples

The most basic examples of topological manifolds:

• Euclidean space $\mathbb{R}^n$ is an $n$-manifold with the single global chart $(\mathbb{R}^n, \mathrm{id})$.
• The $n$-sphere $S^n = \{x \in \mathbb{R}^{n+1} : |x| = 1\}$ is an $n$-manifold covered by two stereographic-projection charts.
• The $n$-torus $T^n = S^1 \times \cdots \times S^1$ is an $n$-manifold. The 2-torus arises, for instance, as the configuration space of a double pendulum.
• Real projective space $\mathbb{RP}^n$ is the set of lines through the origin in $\mathbb{R}^{n+1}$, covered by $n+1$ charts.

2. Smooth Manifolds

2.1. Smooth atlases and smooth structure

A topological manifold becomes a smooth manifold once the transition maps are required to be smooth.

Two smooth atlases are equivalent if their union is again a smooth atlas. Each equivalence class contains a unique maximal representative—the union of all compatible atlases—called a smooth structure on $M$.

The standard smooth structure on $S^n$ is given by the two stereographic-projection charts. If $N = (0,\ldots,0,1)$ is the north pole, the stereographic projection from $N$ sends $(x^1, \ldots, x^{n+1}) \in S^n \setminus \{N\}$ to

The analogous map $\sigma_S$ from the south pole covers $S^n \setminus \{S\}$. On the overlap $S^n \setminus \{N, S\}$ the transition map is $\sigma_S \circ \sigma_N^{-1}(y) = y/|y|^2$, the inversion in the unit sphere, which is $C^\infty$ on $\mathbb{R}^n \setminus \{0\}$.

2.2. Smooth maps and diffeomorphisms

Smooth maps compose: if $F \colon M \to N$ and $G \colon N \to P$ are smooth then so is $G \circ F \colon M \to P$. This makes smooth manifolds into a category, with diffeomorphisms as isomorphisms.

2.3. Morphisms and the categorical structure of manifolds

Smooth manifolds and their maps fit into the framework of category theory. Three nested categories are in play:

The forgetful functor $\mathbf{Diff} \to \mathbf{Top}$ sends a smooth manifold to its underlying topological space and a smooth map to the same function viewed as continuous; it preserves composition and identities, so it is genuinely a functor. The functor $\mathbf{Top} \to \mathbf{Set}$ then forgets the topology. Their composite forgets everything geometric.

The key point is that isomorphisms become strictly weaker as structure is forgotten. A diffeomorphism is an isomorphism in $\mathbf{Diff}$; it is also a homeomorphism (isomorphism in $\mathbf{Top}$) and a bijection (isomorphism in $\mathbf{Set}$), but the converses fail. There exist homeomorphisms that are not diffeomorphisms: the classic example is the identity map from $\mathbb{R}$ with its standard smooth structure to $\mathbb{R}$ with the "cubed" smooth structure $\{(\mathbb{R}, x \mapsto x^{1/3})\}$: continuous and bijective in both directions, but $x \mapsto x^{1/3}$ is not smooth at $0$.

Within Diff the morphisms are graded by the rank of the differential:

The inverse function theorem guarantees that a local diffeomorphism at every point is a local diffeomorphism in the neighbourhood sense (locally invertible with smooth inverse), but need not be globally invertible: the map $t \mapsto (\cos t, \sin t) \colon \mathbb{R} \to S^1$ is a local diffeomorphism everywhere but fails to be injective. A smooth embedding is a global, injective local diffeomorphism onto its image; the image $F(M) \subset N$ is then a submanifold.

2.4. Intrinsic geometry: the manifold as its own ambient space

A decisive conceptual point: the abstract smooth manifold does not live inside anything. Charts provide local coordinate maps into $\mathbb{R}^n$, but once the transition functions are verified to be smooth, the ambient $\mathbb{R}^n$ of each chart is discarded. What remains is a self-contained geometric object: $M$ is its own ambient space.

Every geometric quantity of interest—tangent vectors, differential forms, connections, curvature tensors—is defined purely in terms of the smooth structure and (once a metric is chosen) the Riemannian structure. A sphere knows its own geometry from within: a 2-dimensional geometer confined to $S^2$ can, by measuring distances and angles on the surface, compute its Gaussian curvature $K = 1/r^2$ without ever knowing that $S^2$ is sitting inside $\mathbb{R}^3$.

A smooth manifold can be embedded in Euclidean space; Whitney's theorem guarantees that any compact $n$-manifold embeds smoothly in $\mathbb{R}^{2n}$. Once embedded, a second, extrinsic notion of curvature becomes available. For a hypersurface $\iota \colon M^n \hookrightarrow \mathbb{R}^{n+1}$ with unit normal $\nu$, the second fundamental form is

and the mean curvature $H = \mathrm{tr}_g(\mathrm{II})$ and principal curvatures are extrinsic: they depend on the embedding, not on $M$ alone. One can deform $M$ inside $\mathbb{R}^{n+1}$ to change $H$ while leaving the intrinsic metric $g$ and all Riemannian invariants completely unchanged.

Proof sketch (Brioschi formula). Let $(u, v)$ be local coordinates on $M^2 \subset \mathbb{R}^3$ and write $g = E\,\mathrm{d}u^2 + 2F\,\mathrm{d}u\,\mathrm{d}v + G\,\mathrm{d}v^2$ for the induced metric. The extrinsic definition gives $K = (LN - M^2)/(EG - F^2)$ where $L, M, N$ are the coefficients of the second fundamental form. The content of the theorem is that the numerator $LN - M^2$ and denominator $EG - F^2$ always appear in the combination given by the Brioschi formula:

a formula involving only $E, F, G$ and their first and second partial derivatives. The derivation proceeds by computing $LN - M^2$ directly from the embedding via the Gauss–Weingarten equations and then eliminating all normal-vector components using the identity $|\mathbf{r}_u \times \mathbf{r}_v|^2 = EG - F^2$. The resulting expression depends only on $g_{ij}$ and their derivatives, with no reference to the ambient $\mathbb{R}^3$ or the unit normal. In the modern framework (available in §5), the same conclusion follows cleanly: $K$ equals the sectional curvature $K(\sigma) = g(R(\partial_u,\partial_v)\partial_v, \partial_u)/(EG-F^2)$, and the Riemann tensor is built from Christoffel symbols which depend only on $g_{ij}$ and their first derivatives (see [eq:christoffel]). One can verify $K_{\text{intrinsic}} = K_{\text{extrinsic}}$ explicitly for both the unit sphere and the paraboloid $z = (x^2+y^2)/2$ by direct Christoffel-symbol computation. □

In modern language, $K$ equals the sectional curvature of $(M, g)$ ([sectional-curvature]), which is computed from the Riemann tensor of $g$ alone. Gauss's theorem is remarkable because it says that a quantity defined extrinsically (through the ambient normal and principal curvatures) is secretly intrinsic. Bending a flat sheet of paper into a cylinder preserves $g$ and hence $K = 0$; no amount of bending without stretching can introduce intrinsic curvature.

3. The Tangent Space

3.1. Derivations at a point

To differentiate functions on a manifold one needs a notion of a "direction" at a point that is intrinsic—that is, independent of any embedding into Euclidean space. The derivation definition achieves this.

Proof. That $T_pM$ is a vector space is immediate: sums and scalar multiples of derivations are derivations, verified from the linearity and Leibniz rule.

To determine the dimension, choose a chart $(U, \varphi)$ with $p \in U$ and local coordinates $x^i = r^i \circ \varphi$ (where $r^i \colon \mathbb{R}^n \to \mathbb{R}$ is the $i$-th projection). Define

Each $\partial/\partial x^i|_p$ is a derivation, and the set $\{\partial/\partial x^1|_p, \ldots, \partial/\partial x^n|_p\}$ is linearly independent: if $\sum_i a^i \partial/\partial x^i|_p = 0$ then applying to $x^j$ gives $a^j = 0$.

For spanning, let $v \in T_pM$ be arbitrary. Shrinking $U$ if necessary, identify $p$ with $0 \in \mathbb{R}^n$ via $\varphi$. For any $f \in C^\infty(M)$, write $f \circ \varphi^{-1} = f(0) + \sum_i x^i g_i$ for smooth functions $g_i$ with $g_i(0) = \partial(f \circ \varphi^{-1})/\partial r^i(0)$ (by Taylor's theorem with remainder). Then

where we used $v(1) = 0$ (a standard consequence of the Leibniz rule). Hence $v = \sum_i v(x^i)\,\partial/\partial x^i|_p$, and the coordinates of $v$ in the basis are $v^i = v(x^i)$. □

In local coordinates the differential is represented by the Jacobian matrix $[\partial \hat{F}^j / \partial x^i]$ of the coordinate representation [eq:coord-rep].

3.2. The tangent bundle

The transition maps for the $TM$-atlas are $(\hat x, \hat v) \mapsto (\varphi_\beta \circ \varphi_\alpha^{-1}(\hat x),\; J(\hat x)\, \hat v)$ where $J$ is the Jacobian of the coordinate transition; these are $C^\infty$ since $M$ is smooth.

A vector field on $M$ is a smooth section of $TM$: a smooth map $X \colon M \to TM$ with $\pi \circ X = \mathrm{id}_M$. The space of smooth vector fields is denoted $\Gamma(TM)$ or $\mathfrak{X}(M)$.

3.3. Pushforward and pullback

A smooth map $F \colon M \to N$ induces operations on tangent and cotangent objects, but in opposite directions. This covariant/contravariant split is one of the central organisational principles of differential geometry.

Pushforward. The differential $\mathrm{d}F_p \colon T_pM \to T_{F(p)}N$ of Definition [differential] is the pushforward of tangent vectors at $p$. It sends $v \in T_pM$ to the tangent vector $\mathrm{d}F_p(v) \in T_{F(p)}N$ defined by

In local coordinates it is represented by the Jacobian: if $v = v^i\,\partial/\partial x^i$ then $\mathrm{d}F_p(v) = v^i\,(\partial \hat F^j/\partial x^i)\,\partial/\partial y^j$. The pushforward is covariant: it goes in the same direction as $F$.

Vector fields can be pushed forward only when $F$ is a diffeomorphism (so that $F^{-1}$ exists and the result is again a smooth vector field on $N$). Without invertibility, the pushforward of a vector field is not well-defined globally: different points of $M$ mapping to the same point of $N$ would specify different tangent vectors there.

Cotangent spaces and pullback. The cotangent space at $p$ is the dual vector space

Elements of $T^*_pM$ are covectors (or 1-forms at $p$). The coordinate basis dual to $\{\partial/\partial x^i\}$ is $\{\mathrm{d}x^i\}$, characterised by $\mathrm{d}x^i(\partial/\partial x^j) = \delta^i_j$.

Given $F \colon M \to N$, the pullback of a covector $\omega \in T^*_{F(p)}N$ to $T^*_pM$ is

This is contravariant: it pulls covectors back against the direction of $F$, and—crucially—it requires no invertibility. The pullback is defined for every smooth map. More generally, if $\omega$ is a differential $k$-form on $N$, the pullback $F^*\omega$ is a $k$-form on $M$ defined by

The pullback is functorial: $(G \circ F)^* = F^* \circ G^*$ (note the order reversal), and it commutes with the exterior derivative: $F^*(\mathrm{d}\omega) = \mathrm{d}(F^*\omega)$.

4. Riemannian Manifolds

4.1. The metric tensor

In a local chart $(U, \varphi)$ the metric reads

where Einstein's summation convention is in effect. Positive definiteness means the matrix $[g_{ij}]$ is positive definite at every point.

4.2. Existence of Riemannian metrics

Proof. Let $\{(U_\alpha, \varphi_\alpha)\}$ be a locally finite smooth atlas for $M$ (which exists because $M$ is second countable hence paracompact), and let $\{\psi_\alpha\}$ be a smooth partition of unity subordinate to $\{U_\alpha\}$. For each $\alpha$, pull back the standard Euclidean metric via $\varphi_\alpha$ to define a positive semidefinite form on $U_\alpha$:

Set $g = \sum_\alpha \psi_\alpha\, \tilde g_\alpha$. This is a well-defined smooth symmetric 2-tensor. To verify positive definiteness: for any $p \in M$ and $v \neq 0$, choose $\alpha$ such that $\psi_\alpha(p) > 0$; then $p \in U_\alpha$ and $\mathrm{d}(\varphi_\alpha)_p(v) \neq 0$ (since $\varphi_\alpha$ is a local diffeomorphism), so $\tilde g_\alpha|_p(v, v) > 0$, hence

4.3. Lengths and geodesics

The infimum in the distance formula is achieved (locally) by geodesics: curves that locally minimise length and satisfy, in any chart, the geodesic equation

where $\Gamma^k_{ij}$ are the Christoffel symbols defined in [eq:christoffel] below.

5. Curvature

5.1. Connections

To differentiate vector fields on a manifold one needs a notion of parallel transport, encoded by an affine connection.

5.2. The Levi-Civita connection

Proof. Uniqueness. Suppose $\nabla$ is torsion-free and metric-compatible. Write the metric-compatibility condition [eq:metric-compatible] for all three cyclic permutations of $(X, Y, Z)$:

Form the combination $\text{(I)} + \text{(II)} - \text{(III)}$. The right-hand side becomes:

Now apply torsion-freeness $\nabla_U V - \nabla_V U = [U, V]$ to rewrite three of the six inner products:

Substituting and collecting, four of the six terms pair up and simplify. Specifically, $g(Y, \nabla_X Z)$ and $-g(\nabla_X Z, Y)$ cancel (by symmetry of $g$), and $g(\nabla_Z Y, X)$ and $-g(X, \nabla_Z Y)$ cancel likewise. What remains is:

Rearranging gives the Koszul formula:

Since $g$ is non-degenerate, the value of $g(\nabla_X Y, Z)$ for every $Z$ determines $\nabla_X Y$ uniquely. Thus any two torsion-free metric-compatible connections must agree, establishing uniqueness.

Existence. Define a map $\nabla$ by declaring $\nabla_X Y$ to be the unique vector field whose inner product with every $Z$ equals the right-hand side of [eq:koszul] divided by 2; non-degeneracy of $g$ makes this well defined. Checking the connection axioms (linearity and Leibniz in each argument) is a direct computation from the Koszul formula using the properties of the Lie bracket and $g$. Torsion-freeness follows because the right-hand side of [eq:koszul] is symmetric under $X \leftrightarrow Y$ up to the commutator terms that produce exactly $g([X,Y],Z)$: swapping $X$ and $Y$ changes the sign of $g([X,Y],Z)$ and of $g([X,Z],Y) - g([Y,Z],X)$ in a way that forces $g(\nabla_X Y - \nabla_Y X - [X,Y], Z) = 0$ for all $Z$. Metric compatibility is verified by substituting the Koszul definition back into $X(g(Y,Z)) - g(\nabla_X Y, Z) - g(Y, \nabla_X Z)$ and confirming it vanishes. □

In local coordinates the Levi-Civita connection satisfies $\nabla_{\partial_i}\partial_j = \Gamma^k_{ij}\partial_k$ where the Christoffel symbols are

Derivation. Apply the Koszul formula [eq:koszul] with $X = \partial_i$, $Y = \partial_j$, $Z = \partial_l$. Since coordinate vector fields commute, $[\partial_i, \partial_j] = 0$ and all three bracket terms vanish. The derivative terms become:

The Koszul formula thus gives $2\,g(\nabla_{\partial_i}\partial_j, \partial_l) = \partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}$. Writing $\nabla_{\partial_i}\partial_j = \Gamma^k_{ij}\partial_k$ and using $g(\Gamma^k_{ij}\partial_k, \partial_l) = \Gamma^k_{ij} g_{kl}$:

Contracting both sides with $g^{ml}$ (the inverse metric) and using $g^{ml} g_{kl} = \delta^m_k$ yields the formula with the index $k$ relabelled. The symmetry $\Gamma^k_{ij} = \Gamma^k_{ji}$ is immediate from the formula, confirming torsion-freeness.

These are symmetric in $i, j$ (reflecting torsion-freeness) and encode the geometry of $(M,g)$ completely.

5.3. The Riemann, Ricci, and scalar curvatures

The Riemann tensor measures the failure of covariant differentiation to commute: on flat $\mathbb{R}^n$ (with its standard metric) all Christoffel symbols vanish and $R \equiv 0$. Conversely, $R \equiv 0$ implies $(M, g)$ is locally isometric to Euclidean space.

6. Geometric Flows

Geometric flows are families of Riemannian structures that evolve by PDEs driven by curvature. They are among the most powerful tools in modern geometry.

6.1. Curve shortening flow

The CSF is the gradient flow of the length functional $L(\gamma_t)$: it decreases length as quickly as possible while keeping the curve closed and embedded. The enclosed area $A(t)$ satisfies $\mathrm{d}A/\mathrm{d}t = -2\pi$ by the Gauss-Bonnet theorem, so any solution with initial area $A_0$ must become singular at or before $T = A_0/(2\pi)$.

The theorem is due to Gage and Hamilton ^[1] for the convex case and Grayson ^[2] for the general embedded case. The key tool in Gage-Hamilton is the monotonicity of the isoperimetric ratio $L^2/(4\pi A)$ under CSF, which forces the rescaled curves toward the round circle (where the ratio equals 1). Grayson's contribution was to rule out singularities before convexity is achieved.

A concrete discrete realisation of [eq:csf-eq] is the scheme $p_i \leftarrow p_i + \Delta t\,(\bar p_{i+1} + \bar p_{i-1} - 2\bar p_i)/h_i^2$, where $h_i$ is the mean of the two adjacent chord lengths and bars denote the current iterate. Starting from a radial perturbation of a circle, the higher Fourier modes decay rapidly (mode $m$ decays like $e^{-m^2 t}$ near a circle) and the curve rounds before shrinking to a point.

6.2. Ricci flow and geometrisation

Ricci flow was introduced by Hamilton ^[3] as a tool to deform metrics toward one of constant curvature. Positively curved regions contract, negatively curved regions expand, and the flow drives the metric toward a geometric "equilibrium." Hamilton proved that on a closed 3-manifold of positive Ricci curvature the normalised Ricci flow converges to a metric of constant positive sectional curvature, implying the manifold is diffeomorphic to a spherical space form. He also developed the surgery theory needed to continue the flow past singularities.