Namespaces
Variants
Actions

Difference between revisions of "Harmonic form"

From Encyclopedia of Mathematics
Jump to: navigation, search
(Importing text file)
 
m (tex encoded by computer)
 
Line 1: Line 1:
An exterior [[Differential form|differential form]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464601.png" /> on a Riemannian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464602.png" /> satisfying the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464603.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464604.png" /> is the [[Laplace operator|Laplace operator]] corresponding to the Riemannian metric on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464605.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464606.png" /> is the adjoint of the exterior differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464607.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464608.png" /> has compact support, its harmonicity is equivalent to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h0464609.png" />. The harmonic forms of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646010.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646011.png" /> form a vector space, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646012.png" />, over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646013.png" />. If the Riemannian manifold is compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646014.png" /> is finite-dimensional, being the kernel of the elliptic operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646015.png" />. Since a harmonic form is closed, de Rham's theorem generates a natural mapping of the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646016.png" /> into the real cohomology space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646017.png" /> of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646018.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646019.png" />. It follows from the [[Hodge theorem|Hodge theorem]] that this mapping is an isomorphism. In particular, harmonic functions, i.e. harmonic forms of degree zero, are constant on a connected compact manifold.
+
<!--
 +
h0464601.png
 +
$#A+1 = 73 n = 0
 +
$#C+1 = 73 : ~/encyclopedia/old_files/data/H046/H.0406460 Harmonic form
 +
Automatically converted into TeX, above some diagnostics.
 +
Please remove this comment and the {{TEX|auto}} line below,
 +
if TeX found to be correct.
 +
-->
  
Harmonic forms on a compact Riemannian manifold are invariant with respect to any connected group of isometries of this manifold; for a symmetric space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646020.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646021.png" /> coincides with the space of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646022.png" />-forms which are invariant with respect to the largest connected group of isometries.
+
{{TEX|auto}}
 +
{{TEX|done}}
  
A parallel theory of harmonic forms exists for Hermitian manifolds (cf. [[Hermitian structure|Hermitian structure]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646023.png" />. A harmonic form on a Hermitian manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646024.png" /> is a complex form lying in the kernel of the Laplace–Beltrami operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646025.png" /> (cf. [[Laplace–Beltrami equation|Laplace–Beltrami equation]]). The harmonic forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646026.png" /> constitute the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646027.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646028.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646029.png" /> is compact, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646030.png" /> is finite-dimensional and is naturally isomorphic to the Dolbeault cohomology space. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646031.png" /> is a [[Kähler manifold|Kähler manifold]], these two definitions of harmonic forms are really identical, since <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646032.png" />. In such a case
+
An exterior [[Differential form|differential form]] $  \alpha $
 +
on a Riemannian manifold $  M $
 +
satisfying the equation  $  \Delta \alpha = 0 $,
 +
where  $  \Delta = d \delta + \delta d $
 +
is the [[Laplace operator|Laplace operator]] corresponding to the Riemannian metric on  $  M $
 +
and  $  \delta $
 +
is the adjoint of the exterior differential  $  d $.
 +
If  $  \alpha $
 +
has compact support, its harmonicity is equivalent to  $  d \alpha = \delta \alpha = 0 $.  
 +
The harmonic forms of degree  $  p $
 +
on  $  M $
 +
form a vector space,  $  H  ^ {p} ( M) $,
 +
over the field  $  \mathbf R $.  
 +
If the Riemannian manifold is compact, $  H  ^ {p} ( M) $
 +
is finite-dimensional, being the kernel of the elliptic operator  $  \Delta $.
 +
Since a harmonic form is closed, de Rham's theorem generates a natural mapping of the space  $  H  ^ {p} ( M) $
 +
into the real cohomology space $  H  ^ {p} ( M, \mathbf R ) $
 +
of degree  $  p $
 +
of  $  M $.  
 +
It follows from the [[Hodge theorem|Hodge theorem]] that this mapping is an isomorphism. In particular, harmonic functions, i.e. harmonic forms of degree zero, are constant on a connected compact manifold.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646033.png" /></td> </tr></table>
+
Harmonic forms on a compact Riemannian manifold are invariant with respect to any connected group of isometries of this manifold; for a symmetric space  $  M $
 +
the space  $  H  ^ {p} ( M) $
 +
coincides with the space of  $  p $-
 +
forms which are invariant with respect to the largest connected group of isometries.
 +
 
 +
A parallel theory of harmonic forms exists for Hermitian manifolds (cf. [[Hermitian structure|Hermitian structure]])  $  M $.
 +
A harmonic form on a Hermitian manifold  $  M $
 +
is a complex form lying in the kernel of the Laplace–Beltrami operator  $  \square $(
 +
cf. [[Laplace–Beltrami equation|Laplace–Beltrami equation]]). The harmonic forms of type  $  ( p, q) $
 +
constitute the space  $  H  ^ {p,q} ( M) $
 +
over  $  \mathbf C $.
 +
If  $  M $
 +
is compact,  $  H  ^ {p,q} ( M) $
 +
is finite-dimensional and is naturally isomorphic to the Dolbeault cohomology space. If  $  M $
 +
is a [[Kähler manifold|Kähler manifold]], these two definitions of harmonic forms are really identical, since  $  \square = \overline \square \; = \Delta / 2 $.  
 +
In such a case
 +
 
 +
$$
 +
H ^ {p, q } ( M)  = \
 +
\overline{H}\; {} ^ {q, p } ( M)
 +
$$
  
 
and
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646034.png" /></td> </tr></table>
+
$$
 +
H  ^ {k} ( M) \otimes \mathbf C  = \
 +
\sum _ {p + q = k }
 +
H ^ {p, q } ( M).
 +
$$
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646035.png" /> be the Kähler form on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646036.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646037.png" /> be the operator of interior multiplication by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646038.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646039.png" /> be the operator adjoint to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646040.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646041.png" /> be the space of primitive harmonic forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646042.png" />, i.e. forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646043.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646044.png" />. The following equation is valid for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646046.png" />:
+
Let $  \omega $
 +
be the Kähler form on $  M $,  
 +
let $  L $
 +
be the operator of interior multiplication by $  \omega $,  
 +
let $  \Lambda $
 +
be the operator adjoint to $  L $,  
 +
and let $  H _ {0}  ^ {p,q} ( M) $
 +
be the space of primitive harmonic forms of type $  ( p, q) $,  
 +
i.e. forms $  \alpha \in H  ^ {p,q} ( M) $
 +
for which $  \Lambda \alpha = 0 $.  
 +
The following equation is valid for $  p \geq  q $
 +
and $  p + q \leq  \mathop{\rm dim} _ {\mathbf C}  M $:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646047.png" /></td> </tr></table>
+
$$
 +
H ^ {p, q } ( M)  = \
 +
\sum _ {s = 0 } ^ { q }
 +
L  ^ {s} H _ {0} ^ {p - s, q - s } ( M)  \cong
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646048.png" /></td> </tr></table>
+
$$
 +
\cong  \sum _ {s = 0 } ^ { q }
 +
H _ {0} ^ {p - s, q - s } ( M).
 +
$$
  
For a compact Kähler manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646049.png" /> the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646050.png" /> is identical with the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646051.png" /> of holomorphic forms (cf. [[Holomorphic form|Holomorphic form]]) of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646052.png" />. In particular,
+
For a compact Kähler manifold $  M $
 +
the space $  H ^ {p, 0 } ( M) $
 +
is identical with the space $  \Omega  ^ {p} ( M) $
 +
of holomorphic forms (cf. [[Holomorphic form|Holomorphic form]]) of degree $  p $.  
 +
In particular,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646053.png" /></td> </tr></table>
+
$$
 +
H  ^ {1} ( M) \otimes \mathbf C  = \
 +
\Omega  ^ {1} ( M) + \overline{ {\Omega  ^ {1} ( M) }}\; .
 +
$$
  
 
The study of harmonic functions and forms on Riemann surfaces originates with B. Riemann, whose existence theorems were fully proved at the beginning of the 20th century. The theory of harmonic forms on compact Riemannian manifolds was first presented by W.V.D. Hodge [[#References|[1]]].
 
The study of harmonic functions and forms on Riemann surfaces originates with B. Riemann, whose existence theorems were fully proved at the beginning of the 20th century. The theory of harmonic forms on compact Riemannian manifolds was first presented by W.V.D. Hodge [[#References|[1]]].
  
Various generalizations of the theory of harmonic forms were subsequently given. Let there be given a locally flat (analytic) vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646054.png" /> on a Riemannian (Hermitian) manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646055.png" />, and let there be given a Euclidean (Hermitian) metric on the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646056.png" />. By suitably generalizing the Laplace (Laplace–Beltrami) operator [[#References|[4]]], [[#References|[8]]], it is possible to define the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646057.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646058.png" />) of harmonic forms with values in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646059.png" /> (cf. [[Differential form|Differential form]]). If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646060.png" /> is compact, these spaces are finite-dimensional and isomorphic to the corresponding cohomology spaces of de Rham and Dolbeault, which can in turn be interpreted in terms of sheaf cohomology. In the case of locally flat bundles these cohomology spaces are also closely connected with the cohomology spaces of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646061.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646062.png" /> is not compact, the space of square-integrable harmonic forms is isomorphic to the homology space of the complex of square-integrable forms [[#References|[2]]]. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646063.png" /> is a domain with smooth boundary and compact closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646064.png" /> in a Kähler manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646065.png" />, it is also possible to consider the space of harmonic forms of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646066.png" />, with values in an analytic vector bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646067.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646068.png" />, smooth in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646069.png" /> and continuous on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646070.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646071.png" /> is strictly pseudo-convex, this space is finite-dimensional and is isomorphic to the Dolbeault cohomology space corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646072.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046460/h04646073.png" /> .
+
Various generalizations of the theory of harmonic forms were subsequently given. Let there be given a locally flat (analytic) vector bundle $  E $
 +
on a Riemannian (Hermitian) manifold $  M $,  
 +
and let there be given a Euclidean (Hermitian) metric on the fibres of $  E $.  
 +
By suitably generalizing the Laplace (Laplace–Beltrami) operator [[#References|[4]]], [[#References|[8]]], it is possible to define the spaces $  H  ^ {p} ( E) $(
 +
$  H  ^ {p,q} ( E) $)  
 +
of harmonic forms with values in $  E $(
 +
cf. [[Differential form|Differential form]]). If $  M $
 +
is compact, these spaces are finite-dimensional and isomorphic to the corresponding cohomology spaces of de Rham and Dolbeault, which can in turn be interpreted in terms of sheaf cohomology. In the case of locally flat bundles these cohomology spaces are also closely connected with the cohomology spaces of the group $  \pi _ {1} ( M) $.  
 +
If $  M $
 +
is not compact, the space of square-integrable harmonic forms is isomorphic to the homology space of the complex of square-integrable forms [[#References|[2]]]. If $  M $
 +
is a domain with smooth boundary and compact closure $  \overline{M}\; $
 +
in a Kähler manifold $  \widetilde{M}  $,  
 +
it is also possible to consider the space of harmonic forms of type $  ( p, q) $,  
 +
with values in an analytic vector bundle $  E $
 +
over $  \widetilde{M}  $,  
 +
smooth in $  M $
 +
and continuous on $  \overline{M}\; $.  
 +
If $  M $
 +
is strictly pseudo-convex, this space is finite-dimensional and is isomorphic to the Dolbeault cohomology space corresponding to $  E $
 +
over $  M $.
  
 
Harmonic forms are a powerful tool in the study of the cohomology of real and complex manifolds and of cohomology spaces of discrete groups. The theory of harmonic forms yields fundamental cohomological properties of compact Kähler manifolds and, in particular, of projective algebraic varieties [[#References|[1]]], [[#References|[4]]], [[#References|[5]]]. Harmonic forms can be used to establish a connection between the curvature of a compact Riemannian manifold and the triviality of some of its cohomology groups [[#References|[6]]], [[#References|[7]]]. Similar connections have also been obtained in complex analytic geometry [[#References|[4]]], [[#References|[5]]] and in the theory of discrete transformation groups [[#References|[8]]].
 
Harmonic forms are a powerful tool in the study of the cohomology of real and complex manifolds and of cohomology spaces of discrete groups. The theory of harmonic forms yields fundamental cohomological properties of compact Kähler manifolds and, in particular, of projective algebraic varieties [[#References|[1]]], [[#References|[4]]], [[#References|[5]]]. Harmonic forms can be used to establish a connection between the curvature of a compact Riemannian manifold and the triviality of some of its cohomology groups [[#References|[6]]], [[#References|[7]]]. Similar connections have also been obtained in complex analytic geometry [[#References|[4]]], [[#References|[5]]] and in the theory of discrete transformation groups [[#References|[8]]].

Latest revision as of 19:43, 5 June 2020


An exterior differential form $ \alpha $ on a Riemannian manifold $ M $ satisfying the equation $ \Delta \alpha = 0 $, where $ \Delta = d \delta + \delta d $ is the Laplace operator corresponding to the Riemannian metric on $ M $ and $ \delta $ is the adjoint of the exterior differential $ d $. If $ \alpha $ has compact support, its harmonicity is equivalent to $ d \alpha = \delta \alpha = 0 $. The harmonic forms of degree $ p $ on $ M $ form a vector space, $ H ^ {p} ( M) $, over the field $ \mathbf R $. If the Riemannian manifold is compact, $ H ^ {p} ( M) $ is finite-dimensional, being the kernel of the elliptic operator $ \Delta $. Since a harmonic form is closed, de Rham's theorem generates a natural mapping of the space $ H ^ {p} ( M) $ into the real cohomology space $ H ^ {p} ( M, \mathbf R ) $ of degree $ p $ of $ M $. It follows from the Hodge theorem that this mapping is an isomorphism. In particular, harmonic functions, i.e. harmonic forms of degree zero, are constant on a connected compact manifold.

Harmonic forms on a compact Riemannian manifold are invariant with respect to any connected group of isometries of this manifold; for a symmetric space $ M $ the space $ H ^ {p} ( M) $ coincides with the space of $ p $- forms which are invariant with respect to the largest connected group of isometries.

A parallel theory of harmonic forms exists for Hermitian manifolds (cf. Hermitian structure) $ M $. A harmonic form on a Hermitian manifold $ M $ is a complex form lying in the kernel of the Laplace–Beltrami operator $ \square $( cf. Laplace–Beltrami equation). The harmonic forms of type $ ( p, q) $ constitute the space $ H ^ {p,q} ( M) $ over $ \mathbf C $. If $ M $ is compact, $ H ^ {p,q} ( M) $ is finite-dimensional and is naturally isomorphic to the Dolbeault cohomology space. If $ M $ is a Kähler manifold, these two definitions of harmonic forms are really identical, since $ \square = \overline \square \; = \Delta / 2 $. In such a case

$$ H ^ {p, q } ( M) = \ \overline{H}\; {} ^ {q, p } ( M) $$

and

$$ H ^ {k} ( M) \otimes \mathbf C = \ \sum _ {p + q = k } H ^ {p, q } ( M). $$

Let $ \omega $ be the Kähler form on $ M $, let $ L $ be the operator of interior multiplication by $ \omega $, let $ \Lambda $ be the operator adjoint to $ L $, and let $ H _ {0} ^ {p,q} ( M) $ be the space of primitive harmonic forms of type $ ( p, q) $, i.e. forms $ \alpha \in H ^ {p,q} ( M) $ for which $ \Lambda \alpha = 0 $. The following equation is valid for $ p \geq q $ and $ p + q \leq \mathop{\rm dim} _ {\mathbf C} M $:

$$ H ^ {p, q } ( M) = \ \sum _ {s = 0 } ^ { q } L ^ {s} H _ {0} ^ {p - s, q - s } ( M) \cong $$

$$ \cong \sum _ {s = 0 } ^ { q } H _ {0} ^ {p - s, q - s } ( M). $$

For a compact Kähler manifold $ M $ the space $ H ^ {p, 0 } ( M) $ is identical with the space $ \Omega ^ {p} ( M) $ of holomorphic forms (cf. Holomorphic form) of degree $ p $. In particular,

$$ H ^ {1} ( M) \otimes \mathbf C = \ \Omega ^ {1} ( M) + \overline{ {\Omega ^ {1} ( M) }}\; . $$

The study of harmonic functions and forms on Riemann surfaces originates with B. Riemann, whose existence theorems were fully proved at the beginning of the 20th century. The theory of harmonic forms on compact Riemannian manifolds was first presented by W.V.D. Hodge [1].

Various generalizations of the theory of harmonic forms were subsequently given. Let there be given a locally flat (analytic) vector bundle $ E $ on a Riemannian (Hermitian) manifold $ M $, and let there be given a Euclidean (Hermitian) metric on the fibres of $ E $. By suitably generalizing the Laplace (Laplace–Beltrami) operator [4], [8], it is possible to define the spaces $ H ^ {p} ( E) $( $ H ^ {p,q} ( E) $) of harmonic forms with values in $ E $( cf. Differential form). If $ M $ is compact, these spaces are finite-dimensional and isomorphic to the corresponding cohomology spaces of de Rham and Dolbeault, which can in turn be interpreted in terms of sheaf cohomology. In the case of locally flat bundles these cohomology spaces are also closely connected with the cohomology spaces of the group $ \pi _ {1} ( M) $. If $ M $ is not compact, the space of square-integrable harmonic forms is isomorphic to the homology space of the complex of square-integrable forms [2]. If $ M $ is a domain with smooth boundary and compact closure $ \overline{M}\; $ in a Kähler manifold $ \widetilde{M} $, it is also possible to consider the space of harmonic forms of type $ ( p, q) $, with values in an analytic vector bundle $ E $ over $ \widetilde{M} $, smooth in $ M $ and continuous on $ \overline{M}\; $. If $ M $ is strictly pseudo-convex, this space is finite-dimensional and is isomorphic to the Dolbeault cohomology space corresponding to $ E $ over $ M $.

Harmonic forms are a powerful tool in the study of the cohomology of real and complex manifolds and of cohomology spaces of discrete groups. The theory of harmonic forms yields fundamental cohomological properties of compact Kähler manifolds and, in particular, of projective algebraic varieties [1], [4], [5]. Harmonic forms can be used to establish a connection between the curvature of a compact Riemannian manifold and the triviality of some of its cohomology groups [6], [7]. Similar connections have also been obtained in complex analytic geometry [4], [5] and in the theory of discrete transformation groups [8].

References

[1] W.V.D. Hodge, "The theory and application of harmonic integrals" , Cambridge Univ. Press (1952)
[2] G. de Rham, "Differentiable manifolds" , Springer (1984) (Translated from French)
[3a] L. Schwartz, "Equaciones diferenciales parciales elipticas" , Univ. Nac. Colombia (1973)
[3b] L. Schwartz, "Variedades analiticas complejas" , Univ. Nac. Colombia (1956)
[4] R.O. Wells jr., "Differential analysis on complex manifolds" , Springer (1980)
[5] S.S. Chern, "Complex manifolds without potential theory" , Springer (1979)
[6] S.I. Goldberg, "Curvature and homology" , Acad. Press (1962)
[7] K. Yano, S. Bochner, "Curvature and Betti numbers" , Princeton Univ. Press (1953)
[8] Y. Matsushima, S. Murakami, "On vector bundle valued harmonic forms and automorphic forms on symmetric Riemannian manifolds" Ann. of Math. , 78 (1963) pp. 365–416
[9a] J.J. Kohn, "Harmonic integrals on strongly pseudoconvex manifolds I" Ann. of Math. , 78 (1963) pp. 112–148
[9b] J.J. Kohn, "Harmonic integrals on strongly pseudoconvex manifolds II" Ann. of Math. , 79 (1964) pp. 450–472
How to Cite This Entry:
Harmonic form. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonic_form&oldid=14757
This article was adapted from an original article by A.L. Onishchik (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article