请给出非distribution的tangent subbundle实例，并说明二者核心差异

Great question! Let's start by clearing up terminology to avoid confusion, then walk through a concrete example, and finally break down the key differences between tangent subbundles and distributions.

First, the Standard Definitions #

Tangent Subbundle #

A smooth tangent subbundle ( S ) of a manifold ( M )'s tangent bundle ( TM ) is a subset of ( TM ) where:

• Every fiber ( S_p = S \cap T_pM ) (the part of the subbundle over point ( p )) is a linear subspace of ( T_pM ), and its dimension is the same for all ( p \in M ).
• ( S ) acts like a "smoothly varying family" of subspaces: we can find local coordinate patches where the subbundle looks like ( U \times \mathbb{R}^k ) (for some fixed ( k )), with a smooth map that preserves the linear structure of each fiber.

Distribution #

A smooth ( k )-dimensional distribution ( \mathcal{D} ) on ( M ) is just an assignment of a ( k )-dimensional linear subspace ( \mathcal{D}_p \subset T_pM ) to every ( p \in M ), with the requirement that this assignment is smooth. Formally, this means we can cover ( M ) with open sets where each set has ( k ) smooth vector fields that span the subspace at every point in the set.

Wait a second—by these definitions, every tangent subbundle is a distribution! The local trivializations of the subbundle give us exactly the smooth basis vector fields needed for a distribution. The confusion usually comes when people use "distribution" to mean an integrable distribution (a distribution that satisfies the Frobenius condition, meaning it's tangent to a foliation of the manifold). If that's what you're asking about, we have a perfect example.

Example: A Tangent Subbundle That's Not an Integrable Distribution #

Let's take ( M = \mathbb{R}^3 ) with coordinates ( (x, y, z) ). Define two smooth vector fields:

X = \frac{\partial}{\partial x} + y \frac{\partial}{\partial z}, \quad Y = \frac{\partial}{\partial y}

Now consider the subset ( S \subset T\mathbb{R}^3 ) where each fiber ( S_p ) is the span of ( X(p) ) and ( Y(p) ).

Why ( S ) is a Tangent Subbundle #

• For every point ( p = (x,y,z) ), ( X(p) = (1, 0, y) ) and ( Y(p) = (0, 1, 0) ) are linearly independent, so ( \dim S_p = 2 ) everywhere.
• We can easily make local trivializations: for any open set ( U \subset \mathbb{R}^3 ), map a vector ( aX(p) + bY(p) \in S_p ) to ( (p, (a, b)) \in U \times \mathbb{R}^2 ). This is a smooth, fiber-linear diffeomorphism, so ( S ) checks all the boxes for a tangent subbundle.

Why ( S ) Isn't an Integrable Distribution #

An integrable distribution requires that the Lie bracket of any two local sections stays within the distribution. Let's compute the bracket of ( X ) and ( Y ):

[X, Y] = XY - YX = \frac{\partial}{\partial z}

Take the point ( p = (0, 1, 0) ): ( \frac{\partial}{\partial z}|_p = (0,0,1) ). Is this in ( S_p )? The span of ( X(p) = (1,0,1) ) and ( Y(p) = (0,1,0) ) only includes vectors of the form ( (a, b, a) )—the ( z )-component has to equal the ( x )-component. Since ( (0,0,1) ) doesn't fit this, ( [X,Y] ) isn't a section of ( S ). This violates the Frobenius condition, so ( S ) isn't an integrable distribution.

Core Differences #

If We Stick to Standard Definitions (Distribution = Smooth Fixed-Dimension Subspace Assignment) #

• All tangent subbundles are distributions: The local trivialization of a subbundle gives exactly the smooth basis vector fields required for a distribution.
• Not all distributions are tangent subbundles: Only distributions with constant fiber dimension qualify as subbundles. Singular distributions (where the dimension of ( \mathcal{D}_p ) changes from point to point) can't be subbundles, since subbundles require uniform fiber dimension.

If "Distribution" Means Integrable Distribution #

• Tangent subbundles are a larger class: Every integrable distribution is a tangent subbundle, but many tangent subbundles aren't integrable.
• The key distinction is integrability: Integrable distributions are tangent to a foliation of ( M ) by immersed submanifolds (think of slicing the manifold into lower-dimensional submanifolds that fit together smoothly). Tangent subbundles have no such restriction—they can be "twisted" in a way that prevents this kind of foliation, like our ( \mathbb{R}^3 ) example above.

内容的提问来源于stack exchange，提问作者user365200