{ "schema_version": 1, "kind": "ConceptView", "producer": "concept-view-1", "concept_id": "urn:uuid:89f7a365-87fe-475c-a00b-27b4892ce4c3", "labels": { "zh-CN": "黎曼流形", "en": "Riemannian manifold" }, "alternative_labels": [], "scope_note": "", "identity_status": "active", "identity_revision": "7025519665d02ee0b65ecc5b163d35d61db7c781250a97e9f9cd11f93626d7a3", "occurrences": [ { "concept_id": "urn:uuid:89f7a365-87fe-475c-a00b-27b4892ce4c3", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-16/05-classical-mechanics", "anchor": "first-2bbe90746e718753fb9d-zh", "chapter": "16", "section": "16.5", "offset": 10305, "range": [ 18085, 18329 ], "source_revision": "aac6045c8f792acfb5c9989193835b417610f4c945fe5f70074b9b0c3fb0e939", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "黎曼流形(Riemannian manifold)", "context": { "range": [ 18005, 18464 ], "quote": "而值 0 仅当 ${ \\dot { u } } ^ { i } = 0$ 时才可能，因为约束映射是嵌入且没有临界点。一个处处定义了正定内积的流形 $M$ 称为 黎曼流形(Riemannian manifold)；关于此类流形的进一步讨论将在第 18 章中找到。相关联的对称张量场 $g = g _ { i j } \\ : \\mathrm { d } q ^ { i } \\otimes \\mathrm { d } q ^ { j }$ 称为度量张量。这些评述为以下定义提供了动机。", "quote_sha256": "66643dbc385ca56817d4afa969115217b875c01f4681decfc5a78b4b4656e9b7", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-16/05-classical-mechanics", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-d567fd70c8b3b9591bfe8b07" }, { "concept_id": "urn:uuid:89f7a365-87fe-475c-a00b-27b4892ce4c3", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-16/05-classical-mechanics", "anchor": "first-2bbe90746e718753fb9d-en", "chapter": "16", "section": "16.5", "offset": 13983, "range": [ 21292, 21530 ], "source_revision": "6ba31cc0cea910b2bb1780c9e846e4e265229202d450afd615e6b7053ad533bb", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Riemannian manifold", "context": { "range": [ 21073, 21806 ], "quote": "and the value 0 is only possible if ${ \\dot { u } } ^ { i } = 0$ since the constraint map is an embedding and has no critical points. A manifold $M$ with a positive definite inner product defined everywhere is called a Riemannian manifold; further discussion of such manifolds will be found in Chapter 18. The associated symmetric tensor field $g = g _ { i j } \\ : \\mathrm { d } q ^ { i } \\otimes \\mathrm { d } q ^ { j }$ is called the metric tensor. These remarks serve as motivation for the following definition.", "quote_sha256": "e417d47e3efbdac247f1cbb949e9897b73503c86a57b146cebfdd60604ee7a7c", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "zh/chapter-16/05-classical-mechanics", "lang": "zh-CN", "source_category": "translation", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-21b46ff9e370b6dce5d57d11" }, { "concept_id": "urn:uuid:89f7a365-87fe-475c-a00b-27b4892ce4c3", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "zh-CN", "route": "zh/chapter-18/04-pseudo-riemannian-manifolds", "anchor": "xref-04-pseudo-riemannian-manifolds-1", "chapter": "18", "section": "18.4", "offset": 2732, "range": [ 9270, 9479 ], "source_revision": "1bbb27130bf7d8d0ec56ab51f9406b00c7ca2b06919edbe1a20361105c5aba6d", "source_category": "translation", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "黎曼流形(Riemannian manifold)", "context": { "range": [ 9128, 9480 ], "quote": "但一般而言，除非正交归一 向量场 的所有李括号都消失 $[ e _ { i } , e _ { j } ] = 0$，否则不可能在整个坐标 图上实现 $e _ { i } = \\partial _ { x ^ { i } }$。如果度量张量处处正定，则称 $( M , g )$ 为 黎曼流形(Riemannian manifold)，", "quote_sha256": "abfb254503fc802625b71aeb37c177a7ba60dc6a903175e06ecc09fa3498a53f", "environment": null, "excerpt_omitted": false }, "parallel_readings": [ { "route": "en/chapter-18/04-pseudo-riemannian-manifolds", "lang": "en", "source_category": "original_transcription", "basis": "registered-book-edition-section-language-context; not-concept-equivalence" } ], "id": "occ-67430af22286480215a47afd" }, { "concept_id": "urn:uuid:89f7a365-87fe-475c-a00b-27b4892ce4c3", "book_id": "szekeres", "edition_id": "source-snapshot-bc3404156f6e", "lang": "en", "route": "en/chapter-18/04-pseudo-riemannian-manifolds", "anchor": "xref-04-pseudo-riemannian-manifolds-1", "chapter": "18", "section": "18.4", "offset": 3418, "range": [ 9484, 9687 ], "source_revision": "abe4ca4f42fe5272187e66105b690e8b9f1e858c2725731a33be8d30fbcf1939", "source_category": "original_transcription", "review_status": "unreviewed", "role": "occurrence", "role_basis": { "kind": "explicit-source-uuid-marker" }, "label": "Riemannian manifold", "context": { "range": [ 9248, 9741 ], "quote": "but in general it is not possible to achieve that $e _ { i } = \\partial _ { x ^ { i } }$ over an entire coordinate chart unless all Lie brackets of the orthonormal fields vanish $[ e _ { i } , e _ { j } ] = 0$ . 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