A vector field X on a compact manifold M always generates a complete flow — meaning the integral curves exist for all time t ∈ (-∞, ∞). On a non-compact manifold, this can fail. Why?
AOn non-compact manifolds, vector fields are not smooth, so integral curves are not defined
BIntegral curves on non-compact manifolds can escape to infinity in finite time, preventing extension beyond that time
CNon-compact manifolds do not have tangent bundles, so vector fields cannot be defined
DThe flow equations have no solutions on non-compact manifolds because the ODE existence theorem fails
On a non-compact manifold, integral curves can leave every compact set in finite time — they 'run off to infinity.' For example, the vector field X = x²∂/∂x on ℝ has the integral curve x(t) = 1/(1-t) starting at x=1, which blows up at t=1. On a compact manifold, integral curves have nowhere to escape to, so the ODE existence/uniqueness theorem guarantees the flow extends for all time. This is a consequence of compactness ensuring that any finite-time limit of the integral curve must converge to a point in M.
Question 2 True / False
Vector fields on a smooth manifold form a module over the ring of smooth functions C∞(M), not merely a vector space over ℝ.
TTrue
FFalse
Answer: True
You can multiply a vector field X by a smooth function f to get a new vector field fX, defined by (fX)_p = f(p)·X_p. This operation satisfies the module axioms. The set of vector fields is also a vector space over ℝ (you can add fields and scale by constants), but the module structure over C∞(M) is richer and more useful. This distinction matters: for instance, the C∞(M)-linearity (or lack thereof) of various operations on vector fields distinguishes tensorial operations from non-tensorial ones.
Question 3 Short Answer
Let X = x∂/∂x + y∂/∂y on ℝ². What does the flow of X look like geometrically, and what is the flow map φt(x₀, y₀)?
Think about your answer, then reveal below.
Model answer: The flow is radial expansion/contraction: φt(x₀, y₀) = (eᵗx₀, eᵗy₀). Each point moves radially outward from the origin with exponential speed. The integral curves are rays from the origin (excluding the origin itself, which is a fixed point). The flow dilates distances by a factor of eᵗ.
The system of ODEs is dx/dt = x, dy/dt = y, with solutions x(t) = x₀eᵗ, y(t) = y₀eᵗ. This is the flow of the Euler vector field, which generates scaling transformations. Each φt is a diffeomorphism (in fact, a linear map — multiplication by eᵗ). The fixed point at the origin corresponds to the zero of the vector field.
Question 4 True / False
A smooth function f : M → ℝ is constant along the integral curves of a vector field X if and only if X(f) = 0 everywhere.
TTrue
FFalse
Answer: True
If γ(t) is an integral curve of X, then d/dt f(γ(t)) = X_γ(t)(f). So f is constant along γ if and only if Xf vanishes along γ. If X(f) = 0 everywhere on M, then f is constant on every integral curve. Functions satisfying X(f) = 0 are called first integrals or conservation laws of the vector field. This characterization is fundamental in mechanics: conserved quantities are exactly the functions annihilated by the Hamiltonian vector field.