Lesson 5: Differential Geometry I

Part 2: Tangent Spaces and Vector Fields

The Nature of Vectors on Curved Spaces

When we think about vectors in flat space, we can simply draw arrows from one point to another. However, on a curved surface or manifold, we need to be more careful. We can’t generally compare vectors at different points directly, just as we can’t lay a tangent plane from one point on a sphere flat against a tangent plane at another point. This fundamental insight leads us to develop the concept of tangent spaces.

Let’s organize our exploration using the mnemonic “VECTORS”:

Understanding Tangent Spaces

The tangent space TpM at a point p on a manifold M represents all possible directions in which we can move through p. We can understand this concept through several equivalent definitions:

1. Geometric Definition The tangent space contains all vectors tangent to curves through p:

Consider a curve γ(t) with γ(0) = p

Its velocity vector γ’(0) lies in TpM

All such vectors form the tangent space

2. Algebraic Definition Vectors are derivations on the algebra of functions: $v(fg) = f(p)v(g) + g(p)v(f)$

This shows how vectors act as differential operators.

3. Coordinate Definition In local coordinates (x¹,…,xⁿ):

   $$T_pM = \text{span}\left\{\left.\frac{\partial}{\partial x^i}\right|_p\right\}_{i=1}^n$$

The partial derivatives form a basis.

Vector Fields: Smoothly Varying Directions

A vector field X assigns to each point p ∈ M a tangent vector X(p) ∈ TpM. This assignment must vary smoothly:

1. Local Expression In coordinates: $X = X^i\frac{\partial}{\partial x^i}$

where Xⁱ are smooth functions.

2. Action on Functions A vector field acts on functions: $X(f) = X^i\frac{\partial f}{\partial x^i}$

showing its nature as a derivation.

3. Transformation Law Under coordinate change x → y:

   $$X^i\frac{\partial}{\partial x^i} = X^i\frac{\partial y^j}{\partial x^i}\frac{\partial}{\partial y^j}$$

This ensures coordinate independence.

Physical Examples and Applications

1. Velocity Fields in Fluid Flow The fluid velocity:

   $$\mathbf{v}(x,t) = v^i(x,t)\frac{\partial}{\partial x^i}$$

represents the instantaneous motion at each point.

2. Killing Vector Fields These generate symmetries: $\mathcal{L}_X g = 0$

Example: Rotational symmetry of a sphere.

3. Electromagnetic Potential The vector potential: $\mathbf{A} = A_idx^i$

transforms as a vector field under gauge transformations.

The Flow of Vector Fields

Vector fields generate one-parameter families of diffeomorphisms:

1. Flow Equations $\frac{d\phi_t(p)}{dt} = X(\phi_t(p))$ $\phi_0(p) = p$

These describe the integral curves.

2. Lie Derivative Measures how tensor fields change along flow:

   $$\mathcal{L}*X T = \lim*{t \to 0}\frac{\phi_t^*T - T}{t}$$

Example: For functions: $\mathcal{L}_X f = X(f)$

3. Commutators For vector fields X, Y: $[X,Y] = XY - YX$

Measures non-commutativity of flows.

Frame Fields and Moving Frames

A frame field assigns a basis to each tangent space:

1. Local Frames {eᵢ(p)} where i = 1,…,n forms a basis of TpM at each p.

Properties:

• Smoothly varying

• Linearly independent

• Spans TpM

2. Dual Frames {θⁱ} satisfying: $\theta^i(e_j) = \delta^i_j$

Forms basis for cotangent space.

3. Structure Equations $[e_i,e_j] = c_{ij}^k e_k$

Define how frame vectors commute.

Advanced Concepts: Lie Theory Connection

1. Left-Invariant Vector Fields On Lie groups G: $X_L(gh) = (L_g)_*X_L(h)$

Form basis of Lie algebra.

2. Adjoint Representation $\text{ad}_X Y = [X,Y]$

Connects to Lie algebra structure.

3. Exponential Map $\exp(tX)(p) = \phi_t^X(p)$

Connects local and global structure.

Practical Calculations

1. Spherical Vector Fields On S²: $\frac{\partial}{\partial \theta}, \frac{\partial}{\partial \phi}$

Form coordinate basis.

Killing fields: $\frac{\partial}{\partial \phi}$ (rotation)

Show symmetries.

2. Commutator Calculations For X = x∂/∂y, Y = y∂/∂x:

[X,Y] = xy(∂/∂x - ∂/∂y)

Shows non-commutativity

Represents obstruction to integrability

Common Issues and Their Resolution

1. Coordinate Dependence Problem: Over-relying on coordinates Solution: Focus on geometric meaning Example: Interpret vector fields as flows

2. Global vs. Local Problem: Assuming local properties globalize Solution: Consider topology carefully Example: Hairy ball theorem on S²

3. Basis Independence Problem: Confusing basis-dependent calculations Solution: Use invariant operations Example: Lie derivative definition

Exercises for Understanding

1. Basic Computation Calculate the commutator of vector fields:

   $$X = \frac{\partial}{\partial \theta}, Y = \sin\theta\frac{\partial}{\partial \phi}$$

2. Physical Application Find Killing vector fields for:

   • 2D Euclidean plane

   • Surface of sphere

   • Schwarzschild spacetime

3. Geometric Understanding Prove that any vector field on S² must vanish at some point. Hint: Use index theory.

Looking Ahead

In our next section, we’ll explore differential forms and exterior calculus, which provide the natural language for integration on manifolds. We’ll see how:

• Forms generalize integration

• Exterior derivative unifies grad, curl, div

• Stokes’ theorem becomes geometric

Remember: Vector fields aren’t just mathematical objects — they represent how things can move and change on curved spaces. Understanding them deeply means understanding the possible dynamics in our universe.

Next: Part 3 - Differential Forms and Exterior Calculus