向量丛联络的单位分解构造及推广构造任意联络的可行性问询

Great question! This is one of those classic "local-to-global" workhorses in differential geometry that pops up everywhere, so let's unpack how this construction generalizes.

The two examples you mentioned (vector bundle connections, metric-compatible connections on Riemannian manifolds) share two key prerequisites that make the partition of unity argument work:

• Local existence: We can always find the desired object on small enough open sets (local trivializations for vector bundles, coordinate charts where the metric looks like the Euclidean metric for Riemannian manifolds).
• Linear/affine compatibility: The space of objects we're trying to construct is an affine space (or a linear space), so weighted averages (via partition of unity functions) of local objects will still satisfy the defining properties of the object.

For vector bundles, local trivializations let us use the exterior derivative d as a local connection; for Riemannian manifolds, local coordinates let us use the Euclidean connection (which is metric-compatible), and the partition of unity glues these together while preserving the metric compatibility (since the condition is linear in the connection coefficients).

Yes, this construction generalizes widely, but only when the above two prerequisites hold. Here are concrete cases where it works:

• Hermitian connections on complex vector bundles: Just like Riemannian metrics, Hermitian metrics on complex bundles can be constructed locally (on trivializations) and glued with partition of unity. Then, local Hermitian connections (compatible with the Hermitian metric) can be glued the same way to get a global Hermitian connection.
• Connections with additional linear conditions: Suppose you want a connection on a vector bundle that is compatible with a given bundle map (e.g., a homomorphism between two bundles). If you can find local connections that satisfy this compatibility, the partition of unity will preserve it—since the compatibility condition is linear.
• General "linear" geometric structures: Any structure that is defined by linear conditions, and exists locally, can be glued globally with partition of unity. This includes things like torsion-free connections (though for Riemannian manifolds, torsion-freeness + metric compatibility gives the unique Levi-Civita connection, but the existence still relies on this gluing).

It's important to note the limitations:

• No local existence: If the desired object doesn't exist locally, partition of unity can't create it out of thin air. For example, you can't use this to construct a flat connection on a non-flat vector bundle—flat connections don't exist locally on such bundles, so there's nothing to glue.
• Non-linear conditions: If the defining condition of the object is non-linear, weighted averages won't preserve it. For example, if you wanted a connection with constant curvature (a non-linear condition), gluing local constant-curvature connections with partition of unity would almost certainly result in a connection with non-constant curvature.

The partition of unity construction is a universal tool for local-to-global arguments in differential geometry, especially for linear/affine geometric structures. For connections specifically, as long as you can find local connections that satisfy your desired properties (and those properties are preserved under affine combinations), you can glue them together to get a global connection. This is exactly the same logic as the two examples you started with—you're just applying it to different types of connections or additional constraints.

内容的提问来源于stack exchange，提问作者Doc