"affine differential geometry" meaning in All languages combined

{
  "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.)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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          "name": "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": "quote"
        },
        {
          "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 #92;mathcal#123;O#125;. The structure group in affine differential geometry is the affine group.",
          "type": "quote"
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      ],
      "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",
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      "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."
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  "word": "affine differential geometry"
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{
  "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.)",
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          "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": "quote"
        },
        {
          "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 #92;mathcal#123;O#125;. The structure group in affine differential geometry is the affine group.",
          "type": "quote"
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      ],
      "glosses": [
        "A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations."
      ],
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        "(differential geometry) A type of differential geometry in which the differential invariants studied are invariant under volume-preserving affine transformations."
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