"affine differential geometry" meaning in English

{
  "etymology_text": "The term reflects the categorisation developed by German mathematician Felix Klein for his Erlangen programme (1872, Vergleichende Betrachtungen über neuere geometrische Forschungen), in which he found a useful distinction between projective, affine and Euclidean geometry (in order of increasing restrictiveness). (Riemannian geometry was not initially included.)",
  "head_templates": [
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      "expansion": "affine differential geometry (uncountable)",
      "name": "en-noun"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
  "senses": [
    {
      "categories": [
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          "kind": "other",
          "name": "English entries with incorrect language header",
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            "Entries with incorrect language header",
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          "source": "w"
        },
        {
          "kind": "topical",
          "langcode": "en",
          "name": "Differential geometry",
          "orig": "en:Differential geometry",
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      "examples": [
        {
          "text": "The basic difference between Riemannian and affine differential geometry is that in the affine case we introduce volume forms over a manifold instead of metrics.",
          "type": "example"
        },
        {
          "text": "1999, Alexander I. Bobenko, Wolfgang K. Schief, 5: Discrete Indefinite Affine Spheres, Alexander I. Bobenko, Ruedi Seiler (editors), Discrete Integrable Geometry and Physics, Oxford University Press (Clarendon Press), page 113,\nTzitzeica's classical papers are believed to have initiated a new area in mathematics, namely affine differential geometry. […] The present paper extends the above-mentioned approach to affine differential geometry."
        },
        {
          "ref": "2002, C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, Cambridge University Press, page 88",
          "text": "The Tzitzeica surfaces are the analogues of spheres in affine differential geometry and, indeed, are known as affine spheres or affinsphären [39]. According to Nomizu and Sasaki [227], the origins of affine differential geometry reside in this work of Tzitzeica at the turn of the nineteenth century.",
          "type": "quotation"
        },
        {
          "ref": "2012, Miguel Brozos-Vázquez, Peter B. Gilkey, Stana Nikcevic, Geometric Realizations of Curvature, World Scientific (Imperial College Press), page 89",
          "text": "In Chapter 4, we study questions related to real affine differential geometry. The structure group in Riemannian geometry is the orthogonal group #x5C;mathcal#x7B;O#x7D;. The structure group in affine differential geometry is the affine group.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations."
      ],
      "id": "en-affine_differential_geometry-en-noun-R~oVEWMA",
      "links": [
        [
          "differential geometry",
          "differential geometry"
        ],
        [
          "differential",
          "differential"
        ],
        [
          "invariant",
          "invariant"
        ],
        [
          "affine transformation",
          "affine transformation"
        ]
      ],
      "qualifier": "differential geometry",
      "raw_glosses": [
        "(differential geometry) A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations."
      ],
      "tags": [
        "uncountable"
      ],
      "wikipedia": [
        "Erlangen programme",
        "Felix Klein",
        "affine differential geometry"
      ]
    }
  ],
  "word": "affine differential geometry"
}

{
  "etymology_text": "The term reflects the categorisation developed by German mathematician Felix Klein for his Erlangen programme (1872, Vergleichende Betrachtungen über neuere geometrische Forschungen), in which he found a useful distinction between projective, affine and Euclidean geometry (in order of increasing restrictiveness). (Riemannian geometry was not initially included.)",
  "head_templates": [
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      "expansion": "affine differential geometry (uncountable)",
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  ],
  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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        "English uncountable nouns",
        "en:Differential geometry"
      ],
      "examples": [
        {
          "text": "The basic difference between Riemannian and affine differential geometry is that in the affine case we introduce volume forms over a manifold instead of metrics.",
          "type": "example"
        },
        {
          "text": "1999, Alexander I. Bobenko, Wolfgang K. Schief, 5: Discrete Indefinite Affine Spheres, Alexander I. Bobenko, Ruedi Seiler (editors), Discrete Integrable Geometry and Physics, Oxford University Press (Clarendon Press), page 113,\nTzitzeica's classical papers are believed to have initiated a new area in mathematics, namely affine differential geometry. […] The present paper extends the above-mentioned approach to affine differential geometry."
        },
        {
          "ref": "2002, C. Rogers, W. K. Schief, Bäcklund and Darboux Transformations, Cambridge University Press, page 88",
          "text": "The Tzitzeica surfaces are the analogues of spheres in affine differential geometry and, indeed, are known as affine spheres or affinsphären [39]. According to Nomizu and Sasaki [227], the origins of affine differential geometry reside in this work of Tzitzeica at the turn of the nineteenth century.",
          "type": "quotation"
        },
        {
          "ref": "2012, Miguel Brozos-Vázquez, Peter B. Gilkey, Stana Nikcevic, Geometric Realizations of Curvature, World Scientific (Imperial College Press), page 89",
          "text": "In Chapter 4, we study questions related to real affine differential geometry. The structure group in Riemannian geometry is the orthogonal group #x5C;mathcal#x7B;O#x7D;. The structure group in affine differential geometry is the affine group.",
          "type": "quotation"
        }
      ],
      "glosses": [
        "A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations."
      ],
      "links": [
        [
          "differential geometry",
          "differential geometry"
        ],
        [
          "differential",
          "differential"
        ],
        [
          "invariant",
          "invariant"
        ],
        [
          "affine transformation",
          "affine transformation"
        ]
      ],
      "qualifier": "differential geometry",
      "raw_glosses": [
        "(differential geometry) A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations."
      ],
      "tags": [
        "uncountable"
      ],
      "wikipedia": [
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  "word": "affine differential geometry"
}