Foundations of Differential Geometry

What problem differential geometry solves

Differential geometry addresses a structural gap that appears whenever calculus is applied to spaces that are not canonically identified with ℝⁿ. In elementary multivariable calculus, this gap is hidden by Cartesian coordinates, where points, vectors, and coordinate tuples are silently conflated. Once this identification is removed, it becomes necessary to distinguish the space under study from the numerical labels used to describe it. Differential geometry provides the minimal framework to make this distinction explicit and to explain how notions such as direction, differentiation, and basis arise as constructions rather than axioms.

The core problem is representational rather than computational: how to perform calculus on a space whose points are not themselves tuples of numbers. Coordinates are auxiliary structures, not intrinsic features of space, and must therefore be treated as maps.

Coordinates as maps

A coordinate system is a map

$$F:U\subset R^n\to M,$$

where U is an open subset of ℝⁿ and M is the space of interest. The input consists of numerical labels; the output is a point of the space. Coordinates do not live in M, and points of M are not inherently numerical. Any identification between the two is mediated by F.

Given two coordinate systems

$$F:U\to M,G:V\to M,$$

a change of coordinates is the map

$$G^{-1}\circ F:U\to V,$$

whenever it is defined. The space M is unchanged; only its representation varies.

Coordinate curves and tangent vectors

Fix a coordinate map F and a point u = (u¹,…,uⁿ) ∈ U. For each index i, define the coordinate curve

$${\gamma }_i(t)=F(u^1,\dots ,u^i+t,\dots ,u^n).$$

This is a curve in M passing through the point F(u). Its velocity at t = 0 is

$${\gamma }_i^{\prime }(0)=\frac{\partial F}{\partial u^i}(u).$$

These vectors are tangent to M at F(u). The collection

\left{ \frac{∂F}{∂u¹}(u), …, \frac{∂F}{∂uⁿ}(u) \right}

spans the tangent space at that point. Tangent vectors are therefore defined as derivatives of curves induced by coordinates, not as independent geometric primitives.

Important

Tangent vectors are defined before bases; bases arise only after a choice of coordinates.

Cartesian coordinates reconstructed

Cartesian coordinates correspond to the linear map

$$F(x,y,z)=x\cdot \mathbf{i}+y\cdot \mathbf{j}+z\cdot \mathbf{k}.$$

The coordinate curves are straight lines:

$${\gamma }_x(t)=F(x+t,y,z),{\gamma }_y(t)=F(x,y+t,z),{\gamma }_z(t)=F(x,y,z+t).$$

Differentiating yields

$$\frac{\partial F}{\partial x}=\mathbf{i},\frac{\partial F}{\partial y}=\mathbf{j},\frac{\partial F}{\partial z}=\mathbf{k}.$$

These vectors are constant and orthonormal. All second derivatives vanish:

$$\frac{{\partial }^{2}F}{\partial x^2}=\frac{{\partial }^{2}F}{\partial x\partial y}=\cdots =0.$$

This is why Cartesian coordinates conceal the geometric role of derivatives: the coordinate map is linear, so tangent directions do not vary with position.

First versus second derivatives

First derivatives of a coordinate map encode infinitesimal directions of motion. Given a curve γ(t) in M, its tangent vector is defined as γ’(t). This is a first-order notion. Second derivatives measure how these tangent directions change along curves and therefore encode curvature of the coordinate grid rather than direction itself.

Coordinates stop at first order because their role is to identify directions, not to describe how those directions vary. Higher derivatives become relevant only when differentiating vector fields along curves.

One-dimensional manifolds: the circle

Consider the unit circle S¹ as a geometric space. Introduce an angular coordinate via the map

$$F(\theta )=(\cos \theta ,\sin \theta ).$$

The derivative

$$\frac{dF}{d\theta }=(-\sin \theta ,\cos \theta )$$

defines the unique tangent direction at each point. No global constant tangent vector exists; the tangent direction necessarily depends on θ.

Warning

The position vector F(θ) and the tangent vector dF/dθ are elements of different spaces and must not be identified.

This example shows that coordinate-dependent bases are intrinsic to nontrivial geometry.

Surfaces embedded in ℝ³

A surface embedded in ℝ³ is described by a parametrization

$$F(u,v):U\subset R^2\to R^3.$$

The partial derivatives

$$\frac{\partial F}{\partial u},\frac{\partial F}{\partial v}$$

define two independent tangent directions, and their span defines the tangent plane at each point. Regularity requires these vectors to be linearly independent.

Different parametrizations of the same surface induce different bases of the same tangent plane. The geometric object is the plane itself; the basis is representation-dependent.

Cylindrical coordinates as a chart on ℝ³

Cylindrical coordinates are defined by the map

$$F(r,\theta ,z)=(r\cos \theta ,r\sin \theta ,z).$$

Its partial derivatives are

$$\frac{\partial F}{\partial r}=(\cos \theta ,\sin \theta ,0),$$ $$\frac{\partial F}{\partial \theta }=(-r\sin \theta ,r\cos \theta ,0),$$ $$\frac{\partial F}{\partial z}=(0,0,1).$$

Normalizing yields the unit vectors

$${\mathbf{e}}_r=(\cos \theta ,\sin \theta ,0),{\mathbf{e}}_{\theta }=(-\sin \theta ,\cos \theta ,0),{\mathbf{e}}_z=(0,0,1).$$

Important

Cylindrical coordinates do not introduce new geometry; they introduce a nonlinear coordinate map.

Surfaces r = const are mapped to cylinders because this property is encoded directly in F.

Change of basis on tangent spaces

At a fixed point, different coordinate systems induce different bases of the same tangent space. Relations between bases are linear. For cylindrical coordinates, the relations

$${\mathbf{e}}_r=\cos \theta ,\mathbf{i}+\sin \theta ,\mathbf{j},{\mathbf{e}}_{\theta }=-\sin \theta ,\mathbf{i}+\cos \theta ,\mathbf{j}$$

form a linear system for \mathbf i and \mathbf j. Solving via Gauss–Jordan elimination yields

$$\mathbf{i}=\cos \theta ,{\mathbf{e}}_r-\sin \theta ,{\mathbf{e}}_{\theta },\mathbf{j}=\sin \theta ,{\mathbf{e}}_r+\cos \theta ,{\mathbf{e}}_{\theta }.$$

Linear algebra acts after geometry has identified the relevant tangent space; confusing these stages leads to spurious reasoning.

Scope boundaries

This note deliberately excludes metrics, curvature tensors, connections, and differential forms. Its purpose is foundational: to make explicit how calculus on general spaces is constructed from coordinate maps and their derivatives.