: ''For other inequalities named after Wirtinger, see
Wirtinger's inequality.''
In mathematics, the Wirtinger inequality for 2-forms, named after
Wilhelm Wirtinger
Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.
Biography
He was born at Ybbs on the Danube and studied at the Unive ...
, states that on a
Kähler manifold
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arn ...
, the
exterior th power of the
symplectic form In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form.
A symplectic bilinear form is a mapping that is
; Bilinear: Linear in each argument ...
(Kähler form) , when evaluated on a simple (decomposable) -vector of unit volume, is bounded above by . That is,
:
for any orthonormal vectors . In other words, is a
calibration
In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of kno ...
on . An important corollary of the further characterization of equality is that every complex submanifold of a Kähler manifold is volume minimizing in its homology class.
See also
*
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
*
Gromov's inequality for complex projective space
In Riemannian geometry, Gromov's optimal stable 2- systolic inequality is the inequality
: \mathrm_2^n \leq n!
\;\mathrm_(\mathbb^n),
valid for an arbitrary Riemannian metric on the complex projective space, where the optimal bound is attained
b ...
*
Systolic geometry
Notes
References
*{{cite book, last = Federer, first = Herbert, author-link1=Herbert Federer, title = Geometric measure theory, place= Berlin–Heidelberg–New York, publisher =
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 ...
, series = Die Grundlehren der mathematischen Wissenschaften, volume = 153, year = 1969, isbn = 978-3-540-60656-7, mr=0257325, zbl= 0176.00801 , doi=10.1007/978-3-642-62010-2
Inequalities
Differential geometry
Systolic geometry