In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:
* φ is closed: dφ = 0, where d is the exterior derivative
* for any x ∈ M and any oriented p-dimensional subspace ξ of TxM, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g. Set Gx(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.) Let G(φ) be the union of Gx(φ) for x in M.
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| - Calibrated geometry (en)
- 측정기하학 (ko)
- Калибровочная дифференциальная форма (ru)
- Калібрувальна диференціальна форма (uk)
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| - 미분기하학에서 측정기하학(測定幾何學, 영어: calibrated geometry)은 측정 형식(calibration)이 주어진 매끄러운 다양체를 다루는 분야이다. (ko)
- Калибровочная форма — дифференциальная форма на римановом многообразии. Инструмент в теории минимальных поверхностей позволяющий доказать минимальность площади. (ru)
- Калібрувальна форма — диференціальна форма на рімановому многовиді. Інструмент в теорії мінімальних поверхонь, що дозволяє довести мінімальність площі. (uk)
- In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:
* φ is closed: dφ = 0, where d is the exterior derivative
* for any x ∈ M and any oriented p-dimensional subspace ξ of TxM, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g. Set Gx(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.) Let G(φ) be the union of Gx(φ) for x in M. (en)
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| - In the mathematical field of differential geometry, a calibrated manifold is a Riemannian manifold (M,g) of dimension n equipped with a differential p-form φ (for some 0 ≤ p ≤ n) which is a calibration, meaning that:
* φ is closed: dφ = 0, where d is the exterior derivative
* for any x ∈ M and any oriented p-dimensional subspace ξ of TxM, φ|ξ = λ volξ with λ ≤ 1. Here volξ is the volume form of ξ with respect to g. Set Gx(φ) = { ξ as above : φ|ξ = volξ }. (In order for the theory to be nontrivial, we need Gx(φ) to be nonempty.) Let G(φ) be the union of Gx(φ) for x in M. The theory of calibrations is due to R. Harvey and B. Lawson and others. Much earlier (in 1966) Edmond Bonan introduced G2-manifolds and Spin(7)-manifolds, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifolds were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form. (en)
- 미분기하학에서 측정기하학(測定幾何學, 영어: calibrated geometry)은 측정 형식(calibration)이 주어진 매끄러운 다양체를 다루는 분야이다. (ko)
- Калибровочная форма — дифференциальная форма на римановом многообразии. Инструмент в теории минимальных поверхностей позволяющий доказать минимальность площади. (ru)
- Калібрувальна форма — диференціальна форма на рімановому многовиді. Інструмент в теорії мінімальних поверхонь, що дозволяє довести мінімальність площі. (uk)
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