"Gauss-Codazzi equation" meaning in English

{
  "etymology_text": "Named after Carl Friedrich Gauss and Delfino Codazzi.",
  "forms": [
    {
      "form": "Gauss-Codazzi equations",
      "tags": [
        "plural"
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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      "glosses": [
        "Any of a group of fundamental formulas in Riemannian and pseudo-Riemannian geometry that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold."
      ],
      "id": "en-Gauss-Codazzi_equation-en-noun-mNBX~4vv",
      "links": [
        [
          "geometry",
          "geometry"
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        [
          "submanifold",
          "submanifold"
        ],
        [
          "immersion",
          "immersion"
        ],
        [
          "Riemannian",
          "Riemannian manifold"
        ],
        [
          "pseudo-Riemannian manifold",
          "pseudo-Riemannian manifold"
        ]
      ],
      "raw_glosses": [
        "(geometry) Any of a group of fundamental formulas in Riemannian and pseudo-Riemannian geometry that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold."
      ],
      "topics": [
        "geometry",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "Gauss-Codazzi equation"
}

{
  "etymology_text": "Named after Carl Friedrich Gauss and Delfino Codazzi.",
  "forms": [
    {
      "form": "Gauss-Codazzi equations",
      "tags": [
        "plural"
      ]
    }
  ],
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  "lang_code": "en",
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      "glosses": [
        "Any of a group of fundamental formulas in Riemannian and pseudo-Riemannian geometry that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold."
      ],
      "links": [
        [
          "geometry",
          "geometry"
        ],
        [
          "submanifold",
          "submanifold"
        ],
        [
          "immersion",
          "immersion"
        ],
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          "Riemannian",
          "Riemannian manifold"
        ],
        [
          "pseudo-Riemannian manifold",
          "pseudo-Riemannian manifold"
        ]
      ],
      "raw_glosses": [
        "(geometry) Any of a group of fundamental formulas in Riemannian and pseudo-Riemannian geometry that link together the induced metric and second fundamental form of a submanifold of (or immersion into) a Riemannian or pseudo-Riemannian manifold."
      ],
      "topics": [
        "geometry",
        "mathematics",
        "sciences"
      ]
    }
  ],
  "word": "Gauss-Codazzi equation"
}