{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:346db9a3-44c0-42d4-beda-f834200ff290", "labels": { "zh-CN": "向量丛", "en": "vector bundle" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "98cef015a29ce7ba5a727c42bcd10b38031315c9fbe6ea50c97576bd8f35773b", "occurrences": [ { "concept_id": "urn:uuid:346db9a3-44c0-42d4-beda-f834200ff290", "book_id": "woit-qtgr", "edition_id": "2025-10-20", "lang": "zh-CN", "route": "books/woit-qtgr/2025-10-20/zh/chapter-07/05-the-bloch-sphere", "anchor": "first-997b23b0e8427f44f092-zh", "chapter": "7", "section": "7.5", "offset": 7236, "range": [ 15099, 15406 ], "source_revision": "de552b03ab3055e24dfb26031f138fb19cbe14a7e10ebbc86ec881b4f379f724", "source_category": "machine_draft", "review_status": "machine-draft-not-reviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "向量丛(vector bundle)", "context": { "range": [ 14906, 16191 ], "quote": "\n> [!theorem] 拓展\n>\n> *对每个 $\\mathbf{x}\\in S^2$，方程 [7.5](#equation-7-5) 的解给出 $\\mathcal{H}=\\mathbf{C}^2$ 中的一个一维子空间。随着 $\\mathbf{x}$ 变化，这些子空间构成以球面 $S^2$ 为底空间的[向量丛(vector bundle)](../../../../../concepts/differential-geometry/vector-bundle.md)。对每个 $\\mathbf{x}$ 选定一个解 $u_+(\\mathbf{x})$，便得到该丛的一个[截面(section)](../../../../../concepts/differential-geometry/vector-bundle-section.md)。它可以看作球面上的一种“扭曲”复值函数：普通复值函数在每个点都取值于同一个 $\\mathbf{C}$，而这里的函数在点 $\\mathbf{x}$ 处取值于随 $\\mathbf{x}$ 变化的 $\\mathbf{C}$ 副本。*\n>\n> *这些 $\\mathbf{C}$ 的副本在 $\\mathbf{C}^2$ 中以拓扑非平凡的方式移动，无法彼此连续地等同。这里的向量丛或许是拓扑非平凡向量丛最基本的例子。由于这种拓扑非平凡性，[方程 7.6](#equation-7-6) 的解 $u_+$ 中那样的不连续性不可避免。对于这样的非平凡向量丛，不存在处处非零且连续的截面。*\n", "quote_sha256": "33fdc06cdfd2175b3ea8d69243fd96ab91cab3a4ef9eb4182fc342154b3713ba", "environment": "theorem", "excerpt_omitted": false }, "parallel_readings": [ { "route": "books/woit-qtgr/2025-10-20/en/chapter-07/05-the-bloch-sphere", "lang": "en", "source_category": "machine_draft", "basis": "registered-book-edition-section-language-context; 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