See Klein geometry on Wiktionary
Download JSON data for Klein geometry meaning in All languages combined (6.4kB)
{
"etymology_text": "Named after German mathematician Christian Felix Klein (1849—1925). The concept arose from Klein's Erlangen program (published 1872).",
"forms": [
{
"form": "Klein geometries",
"tags": [
"plural"
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"expansion": "Klein geometry (plural Klein geometries)",
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"lang": "English",
"lang_code": "en",
"pos": "noun",
"related": [
{
"_dis1": "48 47 5",
"word": "Cayley-Klein geometry"
}
],
"senses": [
{
"categories": [
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"kind": "topical",
"langcode": "en",
"name": "Differential geometry",
"orig": "en:Differential geometry",
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{
"text": "Given a Klein geometry (G,H), the group G is called the principal group and G#x2F;H is called the space of the geometry.",
"type": "example"
},
{
"text": "The space of a Klein geometry is a smooth manifold of dimension #x5C;operatorname#x7B;dim#x7D;G-#x5C;operatorname#x7B;dim#x7D;H.",
"type": "example"
},
{
"ref": "1934, American Journal of Mathematics, volume 56, Johns Hopkins University Press, page 153",
"text": "The present paper develops the general theory of non-holonomic geometries as generalizations of Klein geometries starting from a set of fundamental assumptions presented in the form of postulates.",
"type": "quotation"
},
{
"ref": "2006, Luciano Boi, “The Aleph of Space”, in Giandomenico Sica, editor, What is Geometry?, Polimetrica, page 91",
"text": "The kernel of a Klein geometry (G,H) is the largest subgroup K of H that is normal in G. A Klein geometry (G,H) is effective if K#x3D;1 and locally effective if K is discrete. A Klein geometry is geometrically oriented if G is connected.",
"type": "quotation"
},
{
"ref": "2009, Andreas Čap, Jan Slovák, Parabolic Geometries I, American Mathematical Society, page 49",
"text": "A careful geometric study of Klein geometries is available in [Sh97, Chapter 4].\nGiven a Klein geometry (G,H) we may first ask whether all of G is “visible” on G#x2F;H, i.e. whether the action #x5C;mathcal#x7B;l#x7D; of G on G#x2F;H is effective. In this case, we call the Klein geometry effective.",
"type": "quotation"
}
],
"glosses": [
"A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n(more formally) an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected.",
"A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;"
],
"id": "en-Klein_geometry-en-noun-SWsdrDmg",
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"symmetry group",
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"group action",
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"Lie group",
"Lie group"
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"closed",
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"Lie subgroup",
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"left coset space",
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"connected",
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"qualifier": "differential geometry",
"raw_glosses": [
"(differential geometry) A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n"
]
},
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"glosses": [
"A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n(more formally) an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected.",
"an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected."
],
"id": "en-Klein_geometry-en-noun-NnIChz2n",
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[
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"qualifier": "differential geometry; more formally",
"raw_glosses": [
"(differential geometry) A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n"
]
},
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"glosses": [
"The coset space G / H."
],
"id": "en-Klein_geometry-en-noun-daycMMiC",
"raw_glosses": [
"(loosely) The coset space G / H."
],
"tags": [
"broadly"
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}
],
"translations": [
{
"_dis1": "51 48 1",
"code": "de",
"lang": "German",
"sense": "type of geometry",
"tags": [
"feminine"
],
"word": "Kleinsche Geometrie"
}
],
"wikipedia": [
"Erlangen program",
"Felix Klein",
"Klein geometry"
],
"word": "Klein geometry"
}
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"etymology_text": "Named after German mathematician Christian Felix Klein (1849—1925). The concept arose from Klein's Erlangen program (published 1872).",
"forms": [
{
"form": "Klein geometries",
"tags": [
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}
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"lang_code": "en",
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{
"word": "Cayley-Klein geometry"
}
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"senses": [
{
"categories": [
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"en:Differential geometry"
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"examples": [
{
"text": "Given a Klein geometry (G,H), the group G is called the principal group and G#x2F;H is called the space of the geometry.",
"type": "example"
},
{
"text": "The space of a Klein geometry is a smooth manifold of dimension #x5C;operatorname#x7B;dim#x7D;G-#x5C;operatorname#x7B;dim#x7D;H.",
"type": "example"
},
{
"ref": "1934, American Journal of Mathematics, volume 56, Johns Hopkins University Press, page 153",
"text": "The present paper develops the general theory of non-holonomic geometries as generalizations of Klein geometries starting from a set of fundamental assumptions presented in the form of postulates.",
"type": "quotation"
},
{
"ref": "2006, Luciano Boi, “The Aleph of Space”, in Giandomenico Sica, editor, What is Geometry?, Polimetrica, page 91",
"text": "The kernel of a Klein geometry (G,H) is the largest subgroup K of H that is normal in G. A Klein geometry (G,H) is effective if K#x3D;1 and locally effective if K is discrete. A Klein geometry is geometrically oriented if G is connected.",
"type": "quotation"
},
{
"ref": "2009, Andreas Čap, Jan Slovák, Parabolic Geometries I, American Mathematical Society, page 49",
"text": "A careful geometric study of Klein geometries is available in [Sh97, Chapter 4].\nGiven a Klein geometry (G,H) we may first ask whether all of G is “visible” on G#x2F;H, i.e. whether the action #x5C;mathcal#x7B;l#x7D; of G on G#x2F;H is effective. In this case, we call the Klein geometry effective.",
"type": "quotation"
}
],
"glosses": [
"A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n(more formally) an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected.",
"A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;"
],
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"differential geometry",
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"symmetry group"
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"group action"
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"Lie group",
"Lie group"
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"qualifier": "differential geometry",
"raw_glosses": [
"(differential geometry) A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n"
]
},
{
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"glosses": [
"A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n(more formally) an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected.",
"an ordered pair (G, H), where G is a Lie group and H a closed Lie subgroup of G such that the left coset space G / H is connected."
],
"links": [
[
"differential geometry",
"differential geometry"
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"symmetry group"
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"qualifier": "differential geometry; more formally",
"raw_glosses": [
"(differential geometry) A type of geometry (mathematical object representing a space and its spatial relationships); a homogeneous space X together with a symmetry group which represents the group action on X of some Lie group;\n"
]
},
{
"glosses": [
"The coset space G / H."
],
"raw_glosses": [
"(loosely) The coset space G / H."
],
"tags": [
"broadly"
]
}
],
"translations": [
{
"code": "de",
"lang": "German",
"sense": "type of geometry",
"tags": [
"feminine"
],
"word": "Kleinsche Geometrie"
}
],
"wikipedia": [
"Erlangen program",
"Felix Klein",
"Klein geometry"
],
"word": "Klein geometry"
}
This page is a part of the kaikki.org machine-readable All languages combined dictionary. This dictionary is based on structured data extracted on 2024-04-24 from the enwiktionary dump dated 2024-04-21 using wiktextract (82c8ff9 and f4967a5). 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.
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