Differential Geometry: Differentiable Manifolds and Geometric Structures

Differential geometry provides the rigorous mathematical framework for studying spaces that are locally Euclidean but globally curved. By abstracting the concepts of calculus to manifolds, we can describe the geometry of general relativity, the mechanics of constrained systems, and the topology of high-dimensional data.

1. Differentiable Manifolds: The Foundation

A topological manifold $M$ of dimension $n$ is a Hausdorff, second-countable space that is locally homeomorphic to ${\mathbb{R}}^n$. To perform calculus, we require a differentiable structure.

1.1 Charts and Atlases

A coordinate chart is a pair $(U,\varphi )$, where $U\subset M$ is an open set and $\varphi :U\to {\mathbb{R}}^n$ is a homeomorphism. An atlas is a collection of charts $\{(U_{\alpha },{\varphi }_{\alpha })\}$ such that $M={\cup }_{\alpha }{U}_{\alpha }$.

For $M$ to be a differentiable manifold, the transition maps between overlapping charts must be smooth ($C^{\infty }$). Given two charts $(U_{\alpha },{\varphi }_{\alpha })$ and $(U_{\beta },{\varphi }_{\beta })$, the transition map is defined as: ${\varphi }_{\beta \alpha }={\varphi }_{\beta }\circ {\varphi }_{\alpha }^{-1}:{\varphi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\varphi }_{\beta }(U_{\alpha }\cap U_{\beta })$ Since the domain and codomain are subsets of ${\mathbb{R}}^n$, we can apply standard multivariable calculus to demand that ${\varphi }_{\beta \alpha }$ be a diffeomorphism.

2. The Tangent Space $T_{p}M$

At any point $p\in M$, we define a vector space called the tangent space, denoted $T_{p}M$. There are two primary equivalent ways to define this rigorously.

2.1 Equivalence Classes of Curves

Let $\gamma :(-ϵ,ϵ)\to M$ be a smooth curve with $\gamma (0)=p$. Two curves ${\gamma }_1$ and ${\gamma }_2$ are equivalent if their derivatives in a coordinate chart coincide at $t=0$: $(\varphi \circ {\gamma }_1{)}^{\prime }(0)=(\varphi \circ {\gamma }_2{)}^{\prime }(0)$ A tangent vector is an equivalence class of such curves.

2.2 Derivations

Algebraically, a tangent vector $v\in T_{p}M$ is a derivation on the algebra of smooth functions $C^{\infty }(M)$. It is a linear map $v:C^{\infty }(M)\to \mathbb{R}$ satisfying the Leibniz rule: $v(fg)=f(p)v(g)+g(p)v(f)$ In a local coordinate system $\{x^1,\dots ,x^n\}$, the partial derivatives $\{\frac{\partial }{\partial x^1},\dots ,\frac{\partial }{\partial x^n}\}$ form a basis for $T_{p}M$. Any vector $X_p$ can be written as $X^i\frac{\partial }{\partial x^i}$ (using Einstein summation notation).

3. Vector Fields and the Lie Bracket

A vector field $X$ is a smooth assignment of a tangent vector $X_p\in T_{p}M$ to every point $p\in M$. Formally, $X$ is a smooth section of the tangent bundle $TM$.

3.1 The Lie Bracket

The space of smooth vector fields, denoted $\mathfrak{X}(M)$, forms a Lie algebra under the Lie bracket $[X,Y]$. For any $f\in C^{\infty }(M)$: $[X,Y]f=X(Yf)-Y(Xf)$ In local coordinates, if $X=X^i{\partial }_i$ and $Y=Y^j{\partial }_j$, then: $[X,Y]=\left(X^i\frac{\partial Y^j}{\partial x^i}-Y^i\frac{\partial X^j}{\partial x^i}\right)\frac{\partial }{\partial x^j}$ The Lie bracket measures the failure of the flows of $X$ and $Y$ to commute.

4. Differential Forms, Tensors, and Mappings

Geometric objects on manifolds are generalized using tensors.

4.1 Cotangent Space and 1-Forms

The dual space to $T_{p}M$ is the cotangent space $T_p^{\ast }M$. Its elements are linear functionals $\omega :T_{p}M\to \mathbb{R}$. A smooth section of the cotangent bundle $T^{\ast }M$ is a 1-form.

4.2 Pullbacks and Pushforwards

Let $F:M\to N$ be a smooth map between manifolds.

• The pushforward $F_{\ast }:T_{p}M\to T_{F(p)}N$ maps tangent vectors from $M$ to $N$.
• The pullback $F^{\ast }:{\Omega }^k(N)\to {\Omega }^k(M)$ maps differential forms from $N$ to $M$. For a 1-form $\omega$ on $N$: $(F^{\ast }\omega )(v)=\omega (F_{\ast }v)$

5. The Exterior Derivative

The exterior derivative $d$ is the unique linear operator $d:{\Omega }^k(M)\to {\Omega }^{k+1}(M)$ that satisfies:

1. For $f\in {\Omega }^0(M)$, $df$ is the differential of $f$.
2. $d^2=0$ (Poincaré Lemma).
3. $d(\omega \wedge \eta )=d\omega \wedge \eta +(-1{)}^k\omega \wedge d\eta$.

This operator allows us to define De Rham cohomology, linking the analytical properties of forms to the global topology of the manifold.

6. Lie Derivatives and Flows

The Lie derivative ${\mathcal{L}}_X$ captures the “directional derivative” of a tensor field along the flow of a vector field $X$. If ${\theta }_t$ is the flow generated by $X$, then for a vector field $Y$: ${\mathcal{L}}_{X}Y={\lim }_{t\to 0}\frac{({\theta }_{-t}{)}_{\ast }{Y}_{{\theta }_t(p)}-Y_p}{t}$ It is a fundamental result that ${\mathcal{L}}_{X}Y=[X,Y]$.

7. Frobenius Theorem

The Frobenius Theorem provides the condition for a $k$-dimensional sub-bundle $\Delta \subset TM$ (a distribution) to be integrable, meaning there exist submanifolds (leaves) whose tangent spaces are exactly $\Delta$. A distribution $\Delta$ is integrable if and only if it is involutive: $\forall X,Y\in \Delta ⟹[X,Y]\in \Delta$

8. The Exponential Map

On a manifold with a linear connection (usually the Levi-Civita connection of a Riemannian metric), the exponential map ${\exp }_p:T_{p}M\to M$ maps a tangent vector $v$ to the point reached by a geodesic starting at $p$ with initial velocity $v$ after unit time. This allows us to use $T_{p}M$ as a local “linearization” of $M$.

Python Implementation: Computing the Lie Bracket

We use sympy to symbolically compute the Lie bracket of two vector fields in ${\mathbb{R}}^3$ (spherical coordinates).

from sympy import symbols, sin, cos
from sympy.diffgeom import Manifold, Patch, CoordSystem, VectorField

# Define the manifold and coordinate system
M = Manifold('M', 3)
P = Patch('P', M)
Rect = CoordSystem('Rect', P, ['x', 'y', 'z'])
Sph = CoordSystem('Sph', P, ['r', 'theta', 'phi'])

# Define basic coordinates
r, theta, phi = Sph.coord_functions()
e_r, e_theta, e_phi = Sph.base_vectors()

# Define two vector fields
# X = r * d/dr
# Y = d/d_theta
X = r * e_r
Y = e_theta

# Compute Lie Bracket [X, Y]
# In this simple case, [r d/dr, d/d_theta] = -d/d_theta
from sympy.diffgeom import LieBracket
bracket = LieBracket(X, Y)

print(f"Vector Field X: {X}")
print(f"Vector Field Y: {Y}")
print(f"Lie Bracket [X, Y]: {bracket.simplify()}")

Summary Table of Concepts