Curvature forms and Curvature functions for 2-manifolds with boundary
We obrain that any 2-form and any smooth function on a given 2-manifold with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.
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Інститут математики НАН України
2009
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| Цитувати: | Curvature forms and Curvature functions for 2-manifolds with boundary / K. Eftekharinasab // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 484-488. — Бібліогр.: 4 назв. — англ. |
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Eftekharinasab, K. 2010-02-23T14:49:07Z 2010-02-23T14:49:07Z 2009 Curvature forms and Curvature functions for 2-manifolds with boundary / K. Eftekharinasab // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 484-488. — Бібліогр.: 4 назв. — англ. 1815-2910 https://nasplib.isofts.kiev.ua/handle/123456789/6327 We obrain that any 2-form and any smooth function on a given 2-manifold with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively. en Інститут математики НАН України Геометрія, топологія та їх застосування Curvature forms and Curvature functions for 2-manifolds with boundary Article published earlier |
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Curvature forms and Curvature functions for 2-manifolds with boundary |
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Curvature forms and Curvature functions for 2-manifolds with boundary Eftekharinasab, K. Геометрія, топологія та їх застосування |
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Curvature forms and Curvature functions for 2-manifolds with boundary |
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Curvature forms and Curvature functions for 2-manifolds with boundary |
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Curvature forms and Curvature functions for 2-manifolds with boundary |
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Curvature forms and Curvature functions for 2-manifolds with boundary |
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curvature forms and curvature functions for 2-manifolds with boundary |
| author |
Eftekharinasab, K. |
| author_facet |
Eftekharinasab, K. |
| topic |
Геометрія, топологія та їх застосування |
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Геометрія, топологія та їх застосування |
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2009 |
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English |
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Інститут математики НАН України |
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Article |
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We obrain that any 2-form and any smooth function on a given 2-manifold with boundary can be realized as the curvature form and the gaussian curvature function of some Riemmanian metric, respectively.
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1815-2910 |
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https://nasplib.isofts.kiev.ua/handle/123456789/6327 |
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Curvature forms and Curvature functions for 2-manifolds with boundary / K. Eftekharinasab // Збірник праць Інституту математики НАН України. — 2009. — Т. 6, № 2. — С. 484-488. — Бібліогр.: 4 назв. — англ. |
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2025-11-26T01:49:06Z |
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2025-11-26T01:49:06Z |
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1850604601418448896 |
| fulltext |
Çáiðíèê ïðàöüIí-òó ìàòåìàòèêè ÍÀÍ Óêðà¨íè2009, ò.6, �2, 484-488
Kaveh Eftekharinasab
Institute of Mathematics of Ukrainian National Academy of
Sciences
E-mail: kaveh@imath.kiev.ua
Curvature forms and Curvature
functions for 2-manifolds with
boundary
We obrain that any 2-form and any smooth function on a given 2-manifold
with boundary can be realized as the curvature form and the gaussian
curvature function of some Riemmanian metric, respectively.
Keywords: Curvature form,Curvature function, Gauss Bonnet formula, Man-
ifold with boundary.
1. Introduction
For 2-manifolds, possibly, with boundary the classical Gauss
Bonnet formula asserts a relationship between the Euler charac-
teristic of a manifold, its gaussian curvature, and the geodesic
curvature of the boundary. This is the only known obstruction on
a given 2-form on a manifold to be the curvature form of some
Riemmanian metric. Nevertheless, it imposes a constraint on the
sign of a function for being the curvature function of a metric. The
problem of prescribing curvature forms on closed 2-manifolds was
solved by Wallach and Warner [4]. They showed that the Gauss
Bonnet formula is a necessary and sufficient condition on a 2-form
to be a curvature form. Later, the problem of prescribing curva-
ture functions has been studied by some authors and completely
solved for closed manifold by Kazdan and Warner [2]. They proved
that any smooth function which satisfies Gauss Bonnet sign con-
dition, is the gaussian curvature of some Riemmannian metric.
c© Kaveh Eftekharinasab, 2009
Curvature forms and Curvature functions 485
In this paper we deal with 2-manifolds with boundary and the
problems of prescribing curvature forms and curvature functions
which were put to the author by Volodymyr Sharko. In contrast
with the case when manifolds have nonempty boundary no ob-
struction on 2-forms and functions arises. It turns out that any
2-form and smooth function can be realized as the curvature form
and curvature function of a metric respectively, this is a surprising
phenomena.
2. preliminaries and the main results
If we want to study manifolds with boundary we are often faced
with the extension problems, we handle these problems by gluing
manifolds together, providing desired extensions using the elemen-
tary techniques of differential topology. At first, we shall consider
forms then, the same method will be used for functions.
Let M be a connected, compact and oriented 2-manifold with
smooth boundary. Now, glue a 2-disk D2 to M to get a 2-manifold
without boundary M̃ , suitably oriented, joined together along
boundaries. Now we shall have occasion to extend forms from
M to the whole manifold, the existence of extension is an obvious
corollary of the theorem 1.4 [3], that is, if ω1 and ω2 are given
2-forms on M and D2 respectively, (here we just consider 2-forms
but in general it is true for arbitrary forms) which are locally
represented as ω1 = f12 dx
1Λ dx2 and ω2 = g12 dy
1Λdy2 in col-
lar neighborhoods of their boundaries then we can piece together
functions f12 and g12 in bicollar neighborhood as the same as of
the theorem 1.4 [3] and get a smooth function on M̃ and hence
a smooth 2-form ω̃ on M̃ whose restrictions to M and D2 are ω1
and ω2 respectively.
Lemma 1. Let ω be a given 2-form then for any arbitrary nonzero
real number a there exists an extension ω̄ of ω to D2 such that∫
D2 ω̄ = a.
486 K. Eftekharinasab
Proof. Let ω̃ be an arbitrary extension such that
∫
D2 ω̃ 6= 0. We
construct a 2-form ω̄ using bump function such that in an open
neighborhood of the boundary coincides with ω̃ and
∫
D2 ω̄ = a.
Let U be an open neighborhood of the boundary and V be an
open neighborhood of the boundary of the disk D̃2 with the smaller
radius contained in D2. Let fdx1Λdx2 be a local representation
of ω̃ and g be a bump function which equals to the identity in
U and vanishes in V now, put ˜̃ω = g ω̃,
∫
D2
˜̃ω = k 6= 0 and∫
U
˜̃ω = k1,
∫
Ω
˜̃ω = k2 and
∫
fD2
˜̃ω = k3 where Ω is a space between
U and V . Now define a new function h which is equal to the
identity in U and a−k1
k2+k3
g elsewhere. Set ω̄ = h˜̃ω. (Notice that
we always can choose neighborhoods and function g in order to
k2 + k3 6= 0). �
As an evident consequence of this lemma we have the following
corollary.
Corollary 1. For any 2-form ω on M there exists an extension
ω̃ such that ∫
fM
ω̃ = 2πχ(M̃ )
Theorem 1. Let M be a connected, compact and oriented 2-
manifold with smooth boundary then any 2-form ω on M is the
curvature form of some Riemannian metric g on M .
Proof. There exits an extension ω̃ of ω such that
∫
fM
ω̃ = 2πχ(M̃ )
by corollary 1, then employing the theorem of Wallach and Warner
[4] for ω̃, we get a Riemannian metric g̃ on M̃ which its restriction
to M is an expected metric. �
Remark 1. Note that in what, discussed and follows we just
consider manifolds having only one boundary component, but, in
general, when boundary consists of more than one component the
Curvature forms and Curvature functions 487
theorems remain valid, we just need to glue D2 to each component
to get a closed manifold.
Since, we integrate a function, not a 2-form, this fact leads us
to proceed with the same approach, and expect the similar result
for functions, however, we can ask a different question concerns
prescribing gaussian and geodesic curvatures simultaneously, for
example, in [1] the author applies the technique of solving the
Neumann problem on a compact manifold with boundary to the
problem of finding a metric, pointwise conformal to a given metric
with prescribed gaussian curvature and with the prescribed geo-
desic curvature on the boundary when χ(M) ≤ 0. But here our
approach is completely different, indeed our objective is to extend
functions and transfer problems to a closed manifold to avoid dif-
ficulties on boundary. Fortunately, appropriate extensions always
exist
Assume f is a smooth function defined on M the Whitney ex-
tension theorem assures that f can be extended so as to be smooth
throughout M̃ .
Lemma 2. Let f be a smooth function defined on M then there
exists an extension f̃ such that satisfies the sign condition.
Proof. let f̄ be an arbitrary extension which is not zero every-
where, suppose χ(M) > 0 if there exists a point x0 at which
f(x0) > 0 there is nothing to do otherwise multiply f to a func-
tion g, where
g =
{
1, in an open neighborhood of the boundary,
negative, at some point,
fg is a desired extension. If χ(M) < 0 we can modify the extension
likewise. If χ(M) = 0 and f does not vanish identically and does
not change sign, it is strictly positive or negative thus we just
need to multiply it to a function which is equal to the identity in
an open neighborhood of the boundary of D2 and changes sign
elsewhere. �
488 K. Eftekharinasab
Theorem 2. Let M be a compact, connected and oriented 2-
manifold with smooth boundary then any smooth function f is the
gaussian curvature of some Riemmanian metric on M.
Proof. By lemma 2 there exists an extension f̃ of f such that
satisfies the sign condition then by the theorem of Kazdan and
warner [2] there exists a metric on M̃ possesses f̃ as its gaussian
curvature, restriction of the metric toM is an expected metric. �
References
[1] Cherrier P. Problèmes de Neumann non linéaires sur les varietés Rieman-
niennes // J. Funct. Anal. – 1984. – V. 57 – P. 154-206.
[2] Kazdan J. L., Warner F. W. Existence and conformal deformation of met-
rics with prescribed Gaussian and scalar curvatures // Ann. of Math. –
1975. – V. 101, No. 2. – P. 317-331.
[3] Milnor J. Lectures on the h-cobordism theorem. Math. Notes.– Princeton:
Princeton university press, 1965.
[4] Wallach N., Warner F. W. Curvature forms for 2-manifolds // Proc.
Amer. Math. Soc. – 1970. – V. 25. – P. 712–713.
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