See Riemannian manifold in All languages combined, or Wiktionary
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Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold, American Mathematical Society, page 165:", "text": "Further, a much harder theorem, due to J. Nash [31], says that a separable Riemannian manifold of dimension d can be isometrically embedded in #x5C;textstyle#x5C;mathbbRᴺ with N#x5C;le#x7B;#x5C;tfrac 1 2#x7D;d(d#x2B;1)(3d#x2B;11).", "type": "quote" }, { "ref": "2018, John M. Lee, Introduction to Riemannian Manifolds, 2nd edition, Springer, page 55:", "text": "Before we delve into the general theory of Riemannian manifolds, we pause to give it some substance by introducing a variety of \"model Riemannian manifolds\" that should help to motivate the general theory.", "type": "quote" } ], "glosses": [ "A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;" ], "id": "en-Riemannian_manifold-en-noun-pmZAmQRM", "links": [ [ "differential geometry", "differential geometry" ], [ "Riemannian geometry", "Riemannian geometry" ], [ "smooth", "smooth" ], [ "differentiable manifold", "differentiable manifold" ], [ "tangent space", "tangent space" ], [ "positive-definite", "positive-definite" ], [ "inner product", "inner product" ], [ "Riemannian metric", "Riemannian metric" ] ], "qualifier": "differential geometry; Riemannian geometry; differential geometry; Riemannian geometry", "raw_glosses": [ "(differential geometry, Riemannian geometry) A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;" ] }, { "categories": [ { "kind": "topical", "langcode": "en", "name": "Differential geometry", "orig": "en:Differential geometry", "parents": [ "Geometry", "Mathematical analysis", "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w" }, { "_dis": "48 52", "kind": "other", "name": "English entries with incorrect language header", "parents": [ "Entries with incorrect language header", "Entry maintenance" ], "source": "w+disamb" }, { "_dis": "48 52", "kind": "other", "name": "Entries with translation boxes", "parents": [], "source": "w+disamb" }, { "_dis": "47 53", "kind": "other", "name": "Pages with 1 entry", "parents": [], "source": "w+disamb" }, { "_dis": "47 53", "kind": "other", "name": "Pages with entries", "parents": [], "source": "w+disamb" }, { "_dis": "46 54", "kind": "other", "name": "Terms with French translations", "parents": [], "source": "w+disamb" }, { "_dis": "46 54", "kind": "other", "name": "Terms with German translations", "parents": [], "source": "w+disamb" }, { "_dis": "47 53", "kind": "other", "name": "Terms with Italian translations", "parents": [], "source": "w+disamb" }, { "_dis": "46 54", "kind": "other", "name": "Terms with Russian translations", "parents": [], "source": "w+disamb" }, { "_dis": "39 61", "kind": "other", "langcode": "en", "name": "Manifolds", "orig": "en:Manifolds", "parents": [], "source": "w+disamb" }, { "_dis": "47 53", "kind": "topical", "langcode": "en", "name": "Geometry", "orig": "en:Geometry", "parents": [ "Mathematics", "Formal sciences", "Sciences", "All topics", "Fundamental" ], "source": "w+disamb" } ], "derived": [ { "_dis1": "39 61", "word": "pseudo-Riemannian manifold" }, { "_dis1": "39 61", "word": "semi-Riemannian manifold" } ], "glosses": [ "A real, smooth differentiable manifold whose each point has a tangent space equipped with a positive-definite inner product;\n(more formally) an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric.", "an ordered pair (M, g), where M is a real, smooth differentiable manifold and g its Riemannian metric." ], "hypernyms": [ { "_dis1": "39 61", "word": "differentiable manifold" }, { "_dis1": "39 61", "word": "smooth manifold" } ], "id": "en-Riemannian_manifold-en-noun-sH~I9sg3", "links": [ [ "differential geometry", "differential geometry" ], [ "Riemannian geometry", "Riemannian geometry" ], [ "smooth", "smooth" ], [ "differentiable manifold", "differentiable manifold" ], [ "tangent space", "tangent space" ], [ "positive-definite", "positive-definite" ], [ "inner product", "inner product" ], [ "Riemannian metric", "Riemannian metric" ] ], "qualifier": "differential geometry; 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