Calibrated geometry




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-manifold and Spin(7)-manifold, constructed all the parallel forms and showed that those manifolds were Ricci-flat. Quaternion-Kähler manifold were simultaneously studied in 1967 by Edmond Bonan and Vivian Yoh Kraines and they constructed the parallel 4-form.



Calibrated submanifolds


A p-dimensional submanifold Σ of M is said to be a calibrated submanifold with respect to φ (or simply φ-calibrated) if TΣ lies in G(φ).


A famous one line argument shows that calibrated p-submanifolds minimize volume within their homology class. Indeed, suppose that Σ is calibrated, and Σ ′ is a p submanifold in the same homology class. Then


∫ΣvolΣ=∫Σφ=∫Σ′φ≤∫Σ′volΣ′{displaystyle int _{Sigma }mathrm {vol} _{Sigma }=int _{Sigma }varphi =int _{Sigma '}varphi leq int _{Sigma '}mathrm {vol} _{Sigma '}}{displaystyle int _{Sigma }mathrm {vol} _{Sigma }=int _{Sigma }varphi =int _{Sigma '}varphi leq int _{Sigma '}mathrm {vol} _{Sigma '}}

where the first equality holds because Σ is calibrated, the second equality is Stokes' theorem (as φ is closed), and the third inequality holds because φ is a calibration.



Examples



  • On a Kähler manifold, suitably normalized powers of the Kähler form are calibrations, and the calibrated submanifolds are the complex submanifolds. This follows from the Wirtinger inequality.

  • On a Calabi–Yau manifold, the real part of a holomorphic volume form (suitably normalized) is a calibration, and the calibrated submanifolds are special Lagrangian submanifolds.

  • On a G2-manifold, both the 3-form and the Hodge dual 4-form define calibrations. The corresponding calibrated submanifolds are called associative and coassociative submanifolds.

  • On a Spin(7)-manifold, the defining 4-form, known as the Cayley form, is a calibration. The corresponding calibrated submanifolds are called Cayley submanifolds.



References




  • Bonan, Edmond (1965), "Structure presque quaternale sur une variété différentiable", C. R. Acad. Sci. Paris, 261: 5445–5448.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}.


  • Bonan, Edmond (1966), "Sur les variétés riemanniennes à groupe d'holonomie G2 ou Spin(7)", C. R. Acad. Sci. Paris, 262: 127–129.


  • Berger, M. (1970), "Quelques problemes de geometrie Riemannienne ou Deux variations sur les espaces symetriques compacts de rang un", Enseignement Math., 16: 73–96.


  • Brakke, Kenneth A. (1991), "Minimal cones on hypercubes", J. Geom. Anal.: 329–338 (§6.5).


  • Brakke, Kenneth A. (1993), Polyhedral minimal cones in R4.


  • de Rham, Georges (1957–1958), On the Area of Complex Manifolds. Notes for the Seminar on Several Complex Variables, Institute for Advanced Study, Princeton, New Jersey.


  • Federer, Herbert (1965), "Some theorems on integral currents", Transactions of the American Mathematical Society, Transactions of the American Mathematical Society, Vol. 117, 117: 43–67, doi:10.2307/1994196, JSTOR 1994196.


  • Joyce, Dominic D. (2007), Riemannian Holonomy Groups and Calibrated Geometry, Oxford Graduate Texts in Mathematics, Oxford: Oxford University Press, ISBN 978-0-19-921559-1.


  • Harvey, F. Reese (1990), Spinors and Calibrations, Academic Press, ISBN 978-0-12-329650-4.


  • Kraines, Vivian Yoh (1965), "Topology of quaternionic manifolds", Bull. Amer. Math. Soc., 71,3, 1: 526–527, doi:10.1090/s0002-9904-1965-11316-7.


  • Lawlor, Gary (1998), "Proving area minimization by directed slicing", Indiana U. Math. J., 47 (4): 1547–1592, doi:10.1512/iumj.1998.47.1341.


  • Morgan, Frank, Lawlor, Gary (1996), "Curvy slicing proves that triple junctions locally minimize area", J. Diff. Geom., 44: 514–528.


  • Morgan, Frank, Lawlor, Gary (1994), "Paired calibrations applied to soap films, immiscible fluids, and surfaces or networks minimizing other norms", Pac. J. Math., 166: 55–83.


  • McLean, R. C. (1998), "Deformations of calibrated submanifolds", Communications in Analysis and Geometry, 6: 705–747.


  • Morgan, Frank (1988), "Area-minimizing surfaces, faces of Grassmannians, and calibrations", Amer. Math. Monthly, The American Mathematical Monthly, 95 (9): 813–822, doi:10.2307/2322896, JSTOR 2322896.


  • Morgan, Frank (1990), "Calibrations and new singularities in area-minimizing surfaces: a survey In "Variational Methods" (Proc. Conf. Paris, June 1988), (H. Berestycki J.-M. Coron, and I. Ekeland, Eds.)", Prog. Nonlinear Diff. Eqns. Applns, 4: 329–342.


  • Morgan, Frank (2009), Geometric Measure Theory: a Beginner's Guide (4th ed.), London: Academic Press.


  • Thi, Dao Trong (1977), "Minimal real currents on compact Riemannian manifolds", Izv. Akad. Nauk. SSSR Ser. Mat, 41: 807–820.


  • Van, Le Hong (1990), "Relative calibrations and the problem of stability of minimal surfaces", Lecture Notes in Mathematics, New York: Springer-Verlag, 1453: 245–262.


  • Wirtinger, W. (1936), "Eine Determinantenidentität und ihre Anwendung auf analytische Gebilde und Hermitesche Massbestimmung", Monatsh. Math. Phys., 44: 343–365 (§6.5), doi:10.1007/BF01699328.




-

這個網誌中的熱門文章

Tangent Lines Diagram Along Smooth Curve

圖片
7 1 I am trying to replicate a diagram and have a minimal example. I don't know how to add additional tangents between the specified points and preserve smoothness. documentclass{article} usepackage{tikz} usetikzlibrary{calc,intersections} begin{document} begin{tikzpicture} % Axes draw [->, name path=x] (-1,0) -- (11,0) node [right] {$x$}; draw [->] (0,-1) -- (0,6) node [above] {$y$}; % Origin node at (0,0) [below left] {$0$}; % Points coordinate (start) at (1,-0.8); coordinate (c1) at (3,3); coordinate (c2) at (5.5,1.5); coordinate (c3) at (8,4); coordinate (end) at (10.5,-0.8); % show the points foreach n in {start,c1,c2,c3,end} fill [black] (n) circle (2pt) node [below] {}; % join the coordinates draw [thick,name path=curve] (start) to[out=70,in=180] (c1) to[out=0,in=180]
閱讀完整內容

Yusuf al-Mu'taman ibn Hud

圖片
Main article: Taifa of Zaragoza Al-Mu'taman Ruler of Zaragoza Reign 1081–1085 Predecessor Ahmad al-Muqtadir Successor Al-Mustain II Born Zaragoza Died 1085 Full name Yusuf ibn Ahmad al-Mu'taman ibn Hūd House Banu Hud Yusuf ibn Ahmad al-Mu'taman ibn Hūd (Arabic: المؤتمن بالله يوسف إبن أحمد إبن هود ‎, al-Mutaman bi l-Lah , died c. 1085) was the third king of the Banu Hud dynasty who ruled over the Taifa of Zaragoza from 1081 to 1085. Yusuf al-Mutaman reigned during the height of power of Muslim Zaragoza , following the thriving period of his father Ahmad al-Muqtadir. He continued his father's efforts and created around him a court of intellectuals, living in the beautiful palace of Aljafería, nicknamed as "the palace of joy". Al-Mu'taman was also a scholarly king and a patron of science, philosophy and arts. As an example of a wise king, he knew astrology, philosophy, and especially mathematics, a discipline in which
閱讀完整內容

Zucchini

圖片
This article is about the vegetable. For other uses, see Zucchini (disambiguation). Zucchini (courgette) A striped and a uniform color zucchini Genus Cucurbita Species Cucurbita pepo Origin 19th century northern Italy The zucchini ( / z uː ˈ k iː n i / , American English) or courgette ( / k ʊər ˈ ʒ ɛ t / , British English) is a summer squash which can reach nearly 1 metre (100 cm; 39 in) in length, but is usually harvested when still immature at about 15 to 25 cm (6 to 10 in). [1] A zucchini is a thin-skinned cultivar of what in Britain and Ireland is referred to as a marrow. [2] [3] In South Africa, a zucchini is known as a baby marrow. Along with certain other squashes and pumpkins, the zucchini belongs to the species Cucurbita pepo . It can be dark or light green. A related hybrid, the golden zucchini, is a deep yellow or orange color. [4] In a culinary context, the zucchini is treated as a vegetable; it is usually cooked and presented as a savory dish o
閱讀完整內容