See Riemannian manifold in All languages combined, or Wiktionary
{
"etymology_text": "Named after German mathematician Bernhard Riemann (1826–1866). See also Riemannian.",
"forms": [
{
"form": "Riemannian manifolds",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Riemannian manifold (plural Riemannian manifolds)",
"name": "en-noun"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"senses": [
{
"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": "46 54",
"kind": "topical",
"langcode": "en",
"name": "Geometry",
"orig": "en:Geometry",
"parents": [
"Mathematics",
"Formal sciences",
"Sciences",
"All topics",
"Fundamental"
],
"source": "w+disamb"
}
],
"examples": [
{
"text": "By definition, a Riemannian manifold M has at each point p a tangent space T#x5F;pM equipped with a positive-definite inner product, g#x5F;p; information about these inner products is encoded in the Riemannian metric tensor, g.",
"type": "example"
},
{
"ref": "1984, Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, page 55:",
"text": "In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded.",
"type": "quote"
},
{
"ref": "2000, Daniel W. 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": "46 54",
"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; Riemannian geometry; more formally",
"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;"
],
"related": [
{
"_dis1": "39 61",
"word": "differentiable manifold"
},
{
"_dis1": "39 61",
"alt": "= Riemannian metric tensor",
"word": "Riemannian metric"
}
],
"synonyms": [
{
"_dis1": "39 61",
"word": "Riemannian space"
}
]
}
],
"translations": [
{
"_dis1": "47 53",
"code": "fr",
"lang": "French",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"feminine"
],
"word": "variété riemannienne"
},
{
"_dis1": "47 53",
"code": "de",
"lang": "German",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"feminine"
],
"word": "riemannsche Mannigfaltigkeit"
},
{
"_dis1": "47 53",
"code": "de",
"lang": "German",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"masculine"
],
"word": "riemannscher Raum"
},
{
"_dis1": "47 53",
"code": "it",
"lang": "Italian",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"feminine"
],
"word": "varietà riemanniana"
},
{
"_dis1": "47 53",
"code": "ru",
"lang": "Russian",
"roman": "rímanovo mnogoobrázije",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"neuter"
],
"word": "ри́маново многообра́зие"
}
],
"wikipedia": [
"Bernhard Riemann",
"Riemannian manifold"
],
"word": "Riemannian manifold"
}
{
"categories": [
"English countable nouns",
"English entries with incorrect language header",
"English eponyms",
"English lemmas",
"English multiword terms",
"English nouns",
"Entries with translation boxes",
"Pages with 1 entry",
"Pages with entries",
"Terms with French translations",
"Terms with German translations",
"Terms with Italian translations",
"Terms with Russian translations",
"en:Geometry",
"en:Manifolds"
],
"derived": [
{
"word": "pseudo-Riemannian manifold"
},
{
"word": "semi-Riemannian manifold"
}
],
"etymology_text": "Named after German mathematician Bernhard Riemann (1826–1866). See also Riemannian.",
"forms": [
{
"form": "Riemannian manifolds",
"tags": [
"plural"
]
}
],
"head_templates": [
{
"args": {},
"expansion": "Riemannian manifold (plural Riemannian manifolds)",
"name": "en-noun"
}
],
"hypernyms": [
{
"word": "differentiable manifold"
},
{
"word": "smooth manifold"
}
],
"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"word": "differentiable manifold"
},
{
"alt": "= Riemannian metric tensor",
"word": "Riemannian metric"
}
],
"senses": [
{
"categories": [
"English terms with quotations",
"English terms with usage examples",
"en:Differential geometry"
],
"examples": [
{
"text": "By definition, a Riemannian manifold M has at each point p a tangent space T#x5F;pM equipped with a positive-definite inner product, g#x5F;p; information about these inner products is encoded in the Riemannian metric tensor, g.",
"type": "example"
},
{
"ref": "1984, Isaac Chavel, Eigenvalues in Riemannian Geometry, Academic Press, page 55:",
"text": "In this chapter we extend the study of eigenvalues to Riemannian manifolds whose curvature may not be constant, but is, nevertheless, bounded.",
"type": "quote"
},
{
"ref": "2000, Daniel W. 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;"
],
"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": [
"English terms with quotations",
"English terms with usage examples",
"en:Differential geometry"
],
"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."
],
"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; more formally",
"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;"
]
}
],
"synonyms": [
{
"word": "Riemannian space"
}
],
"translations": [
{
"code": "fr",
"lang": "French",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"feminine"
],
"word": "variété riemannienne"
},
{
"code": "de",
"lang": "German",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"feminine"
],
"word": "riemannsche Mannigfaltigkeit"
},
{
"code": "de",
"lang": "German",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"masculine"
],
"word": "riemannscher Raum"
},
{
"code": "it",
"lang": "Italian",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"feminine"
],
"word": "varietà riemanniana"
},
{
"code": "ru",
"lang": "Russian",
"roman": "rímanovo mnogoobrázije",
"sense": "real, smooth differentiable manifold equipped with a Riemannian metric",
"tags": [
"neuter"
],
"word": "ри́маново многообра́зие"
}
],
"wikipedia": [
"Bernhard Riemann",
"Riemannian manifold"
],
"word": "Riemannian manifold"
}
Download raw JSONL data for Riemannian manifold meaning in English (5.3kB)
This page is a part of the kaikki.org machine-readable English dictionary. This dictionary is based on structured data extracted on 2024-12-01 from the enwiktionary dump dated 2024-11-21 using wiktextract (95d2be1 and 64224ec). The data shown on this site has been post-processed and various details (e.g., extra categories) removed, some information disambiguated, and additional data merged from other sources. See the raw data download page for the unprocessed wiktextract data.
If you use this data in academic research, please cite Tatu Ylonen: Wiktextract: Wiktionary as Machine-Readable Structured Data, Proceedings of the 13th Conference on Language Resources and Evaluation (LREC), pp. 1317-1325, Marseille, 20-25 June 2022. Linking to the relevant page(s) under https://kaikki.org would also be greatly appreciated.