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Ph.D. (mathematics)
Irina Gelbukh. Structure of a Morse form foliation on a closed surface in terms of genus. Differential Geometry and its Applications, 29:473–492, 2011.
Erratum: In Corollary 25, "non-planar a compact singular leaf" should read "a non-planar compact singular leaf."
Abstract: We study the geometry of compact singular leaves $\gamma$ and minimal components $C^{min}$ of the foliation $F$ of a Morse form $\omega$ on $M^2_g$ in terms of genus $g(\cdot)$. We show that $c(\omega) + \sum_\gamma g(V(\gamma)) + g(\bigcup \overline {C^{min}})=g$, where $c(\omega)$ is the number of homologically independent compact leaves and $V(\cdot)$ is a small closed tubular neighborhood. This allows us to prove a criterion for compactness of the singular foliation $\overline F$, to estimate the number of its minimal components, and to give an upper bound on the rank $rk \omega$, in terms of genus.
Key words: Morse form foliation; minimal component; compact singular leave; genus; isotropic subgroup
MSC 2000: 57R30, 58K65
JCR impact factor: 0.669 (2009)
This site provides a preprint version of the paper. Final version on the journal's site: DOI 10.1016/j.difgeo.2011.04.029
Irina Gelbukh. On collinear closed one-forms. Bulletin of the Australian Mathematical Society, in press.
Abstract: We study one-forms with zero wedge-product, which we call collinear, and their foliations. We characterise the set of forms that define a given foliation, with special attention to closed forms and forms with small singular sets. We apply the notion of collinearity to give a criterion for existence of a compact leaf and to study homological properties of compact leaves.
Key words: differential one-form; singular set; foliation; compact leaves; cup-product
MSC 2000: 57R30, 58A10
JCR impact factor: 0.302 (2009)
Irina Gelbukh. The number of split points of a Morse form and the structure of its foliation. Mathematica Slovaca, in press.
Abstract: Sharp bounds are given that connect split points---conic singularities of a special type---of a Morse form with the global structure of its foliation.
Key words: Morse form; singularities; foliation; foliation graph
MSC 2000: 57R30, 58K65
JCR impact factor: 0.308 (2009)
This site provides a preprint version of the paper.
Irina Gelbukh. The number of minimal components and homologically independent compact leaves of a weakly generic Morse form on a closed surface. Rocky Mountains Journal of Mathematics, to appear.
Abstract: On a closed orientable surface $M^2_g$ of genus $g$, we consider the foliation of a weakly generic Morse form $\omega$ on $M^2_g$ and show that for such forms $c(\omega) + m(\omega) = g - 1/2 k(\omega)$, where $c(\omega)$ is the number of homologically independent compact leaves of the foliation, $m(\omega)$ the number of its minimal components, and $k(\omega)$ the total number of singularities of $\omega$ surrounded by a minimal component. We also give lower bounds on $m(\omega)$ in terms of $k(\omega)$ and the form rank $rk \omega$ or the structure of $ker [\omega]$, where $[\omega]$ is the integration map.
Key words: Surface; Morse form foliation; number of minimal components
MSC 2000: 57R30, 58K65
JCR impact factor: 0.260 (2009)
This site provides a preprint version of the paper.
Irina Gelbukh. On compact leaves of a Morse form foliation. Publicationes Mathematicae-Debrecen, 78(1):37–48, 2011.
Abstract: On a compact oriented manifold without boundary, we consider a closed 1-form with singularities of Morse type, called Morse form. We give criteria for the foliation defined by this form to have a compact leaf, to have $k$ homologically independent compact leaves, and to have no minimal components.
Key words: Morse form foliation, compact leaves, collinear 1-forms, form rank
MSC 2000: 57R30, 58K65
JCR impact factor: 0.646 (2009)
This site provides a preprint version of the paper. Version "in print". Final version on the journal's website; DOI 10.5486/PMD.2011.4369
Irina Gelbukh. Ranks of collinear Morse forms. Journal of Geometry and Physics, 61(2):425–435, 2011.
Erratum: Section 1, paragraph 3: "in general relativity and quantum cosmology" should read "in general relativity".
Abstract: On a smooth closed $n$-manifold, we consider Morse forms with wedge-product zero; we call such forms collinear. This is an equivalence relation. Collinearity classes are classified by the underlying foliation; so, in other words, we study the set of Morse forms that define the same foliation. We describe the set of the ranks of such forms and show how it is related to the structure of the foliation and the manifold.
Key words: Collinear Morse forms, Morse form foliation, form's rank, minimal components, compact leaves
MSC 2000: 57R30, 58K65
JCR impact factor: 0.714 (2009)
This site provides a preprint version of the paper. Final version on the journal's website; DOI 10.1016/j.geomphys.2010.10.010
Irina Gelbukh. Number of minimal components and homologically independent compact leaves for a Morse form foliation. Studia Scientiarum Mathematicarum Hungarica, 46(4):547–557, 2009.
Erratum: page 551, "the next $k_1$ factors correspond to the set of weakly complete minimal components," should read "<...> not weakly complete <...>".
Abstract: The numbers $m(\omega)$ of minimal components and $c(\omega)$ of homologically independent compact leaves of the foliation of a Morse form $\omega$ on a connected smooth closed oriented manifold $M$ are studied in terms of the first non-commutative Betti number $b_1'$. A sharp estimate $0 \le m(\omega) + c(\omega) \le b_1'$ is given. It is shown that all values of $m(\omega) + c(\omega)$, and in some cases all combinations of $m(\omega)$ and $c(\omega)$ with this condition, are reached on a given $M$. The corresponding issues are also studied in the classes of generic forms and compactifiable foliations.
Key words: Morse form foliation, minimal components, compact leaves
MSC 2000: 57R30, 58K65
JCR impact factor: 0.229 (2009)
This site provides a preprint version of the paper. Final version on the journal's website; DOI 10.1556/SScMath.2009.1108.
Irina Gelbukh. On the structure of a Morse form foliation. Czechoslovak Mathematical Journal, 59(1):207–220, 2009.
Abstract: The foliation of a Morse form $\omega$ on a closed manifold $M$ is considered. Its maximal components (cylinders formed by compact leaves) form the foliation graph; the cycle rank of this graph is calculated. The number of minimal and maximal components is estimated in terms of characteristics of $M$ and $\omega$. Conditions for the presence of minimal components and homologically non-trivial compact leaves are given in terms of $rk \omega$ and $Sing \omega$. The set of the ranks of all forms defining a given foliation without minimal components is described. It is shown that if $\omega$ has more centers than conic singularities then $b_1(M) = 0$ and thus the foliation has no minimal components and homologically non-trivial compact leaves, its foliation graph being a tree.
Key words: number of minimal components, number of maximal components, compact leaves, foliation graph, rank of a form
MSC 2000: 57R30, 58K65
JCR impact factor: 0.306 (2009)
This site provides a draft version for preview purposes. Final version on the journal's site and final version on Springer's site.
Irina Gelbukh. Presence of minimal components in a Morse form foliation. Differential Geometry and its Applications, 22(2):189–198, 2005; Zbl 1070.57016.
Abstract: Conditions and a criterion for the presence of minimal components in the foliation of a Morse form $\omega$ on a smooth closed oriented manifold $M$ are given in terms of (1) the maximum rank of a subgroup in $H^1(M,Z)$ with trivial cup-product, (2) $ker [\omega]$, and (3) $rk \omega = rk im [\omega]$, where $[\omega]$ is the integration map.
Key words: Morse form foliation, minimal components, form rank, cup-product
MSC 2000: 57R30, 58K65
JCR impact factor: 0.391 (2005), 0.669 (2009)
This site provides a draft version for preview purposes. Final version on Elsevier's site.
I.A. Mel'nikova. Maximal isotropic subspaces of
skew-symmetric bilinear mapping. Mosc. Univ. Math. Bull.
54(4):1–3,
1999;
Zbl 0957.57018.
Translated
from: И.А. Мельникова.
Максимальные изотропные подпространства кососимметрического билинейного
отображения.
Вестник МГУ, Сер. матем., мех. 4:3–5, 1999.
Erratum: The paper deals with compactifiable foliations and not compact.
Zentralblatt review: The paper continues the author’s investigations [I. A. Mel’nikova, Math. Notes 58, No. 6, 1302-1305 (1995); translation from Mat. Zametki 58, No. 6, 872-877 (1995; Zbl 0857.57030); Russ. Math. Surv. 50, No. 2, 444-445 (1995); translation from Usp. Mat. Nauk 50, No. 3, 217-218 (1995; Zbl 0859.58005)] in which the compactness problem for the Morse form foliation on a closed manifold $M^n$ is considered. The author discusses the problem of calculation of the maximal isotropic subgroup in $H_{n-1}(M)$ with respect to the operation of intersection of homology classes. The upper and lower estimates are established and some examples are considered when $M = T^n$ is an $n$-dimensional torus and $M = M^2_g$.
MSC: 57R30, 57M07, 54H10 (according to Zentralblatt)
I.A. Mel'nikova. Noncompact leaves of foliations of
Morse forms. Math. Notes 63(6):760–763,
June 1998;
Zbl 0917.57022.
Translated from: И.А. Мельникова.
Некомпактные слои слоения морсовской формы. Матем.
заметки 63(6):862–865, 1998.
Erratum: The paper deals with (non-)compactifiable leaves and not (non-)compact.
Abstract: In this paper foliations determined by Morse forms on compact manifolds are considered. An inequality involving the number of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form is obtained.
Keywords: Morse forms, noncompact leaves of foliations, two-dimensional manifolds
Zentralblatt review: Let $M$ be a compact connected oriented manifold of dimension $n$ with a closed 1-form $\omega$ having only Morse singularities (Morse form). Let $F_\omega$ be a foliation with singularities on $M$ and $[\gamma]$ the homology class of a nonsingular compact leaf $\gamma \in F_\omega$. The image of the set of nonsingular compact leaves generates a subgroup $H_\omega$ in $H_(n-1)(M)$. By $\Omega_i$ denote the set of singular points of index $i$. In this note an inequality involving the number $s$ of connected components of the set formed by noncompact leaves, the number of homologically independent compact leaves, and the number of singular points of the corresponding Morse form $\omega$ is obtained. Theorem. The following inequality holds: $r_k H_\omega + s \le 1/2 (|\Omega_1| - |\Omega_0|) + 1$. P. Arnoux and G. Levitt [Invent. Math. 84, 141–156 (1986; Zbl 0561.58024)] obtained an estimate of $s$ in terms of characteristic of $M$: $s \le 1/2 b_1(M)$. These two estimates coincide for $n = 2$, and they are independent in the case $n > 2$. The method is based on some results of graph theory.
Keywords: singular points of a foliation (according to Zentralblatt)
MSC: 57R30, 53C12, 57R20 (according to Zentralblatt)
JCR impact factor: 0.157 (2001), 0.337 (2009)
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I.A. Mel'nikova. Properties of Morse forms that determine compact foliations on $M_g^2$. Math. Notes 60(6):714–716, 1996; Zbl 0898.57012.
Erratum: The paper deals with compactifiable foliations and not compact.
Abstract: In [1, 2] P. Arnoux and G. Levitt showed that the topology of the foliation of a Morse form $\omega$ on a compact manifold is closely related to the structure of the integration mapping $[\omega]: H_1(M) \to R$. In this paper we consider the foliation of a Morse form on a two-dimensional manifold $M_g^2$. We study the relationship of the subgroup $Ker [\omega] \subset H_1(M_g^2)$ with the topology of the foliation. We consider the structure of the subgroup $Ker [\omega]$ for a compact foliation and prove a criterion for the compactness of a foliation.
Keywords: two-dimensional manifold, foliation, Morse form, integration over cycles.
Zentralblatt review: P. Arnoux and G. Levitt [Invent. Math. 84, 141–156 (1986; Zbl 0561.58024); ibid. 88, 635–667 (1987; Zbl 0594.57014)] showed that the topology of the foliation of a Morse form $\omega$ on a compact manifold is closely related to the structure of the integration mapping $[\omega]: H_1(M) \to \mathbb{R}$. In this paper, the author considers the foliation of a Morse form on a two-dimensional manifold $M_g^2$. He studies the relationship of the subgroup $Ker [\omega] \subset H_1(M_g^2)$ with the topology of the foliation, considers the structure of the subgroup $Ker [\omega]$ for a compact foliation and proves a criterion for the compactness of a foliation.
Keywords
: integration over cycles; Morse form; two-dimensional manifold; compact foliation (according to Zentralblatt)MSC: 57R30, 58C25, 58K99 (according to Zentralblatt)
JCR impact factor: 0.157 (2001), 0.337 (2009)
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Russian: info and full text on the journal's site (open access). The title above links to a locally mirrored file.
I.A. Mel'nikova. Singular points of a Morsian form and foliations.
Mosc. Univ. Math. Bull.
51(4):33–36,
1996;
Zbl 0914.58006.
Translated from: И.А. Мельникова.
Особые точки морсовской формы и слоения.
Вестник МГУ, Сер. матем., мех. 4:37–40,
1996.
Errata: The title should have been translated as "Singularities of a Morse form and foliation". Where the paper mentions the irrationality degree $dirr$, it would be more common to refer to the form rank: $dirr\omega = rk\omega - 1$. The paper deals with compactifiable foliations and not compact.
Zentralblatt review (I had to correct their translation of terminology): Let $M$ be a smooth compact connected orientable $n$-dimensional manifold on which a 1-form $\omega$ with Morse singularities is defined. On the manifold $M$, a foliation with singularities $F_\omega$ is defined. The irrationality degree of the form $\omega$ is determined by $dirr \omega = rk_Q \{\int_{z_1}\omega, ..., \int_{z_m}\omega\} - 1$, where $z_1, ..., z_m$ is the basis in $H_1(M)$. It is proved that in the case of a compact foliation, the irrationality degree of the form and the number of homologically independent leaves are determined by the difference of the numbers of singularities of index 0 and 1.
Keywords: Morse forms; foliation; singular points (according to Zentralblatt)
MSC 2000: 58E05 22E70 (according to Zentralblatt)
I.A. Mel'nikova. A test for compactness of a
foliation.
Math. Notes 58(6):1302–1305, 1995;
Zbl 0857.57030.
Translated from: И.А. Мельникова.
Признак компактности слоения.
Матем. заметки 58(6):872–877, 1995.
Erratum: The paper deals with compactifiable foliations and not compact.
Abstract: We investigate foliations on smooth manifolds that are determined by a closed 1-form with Morse singularities. We introduce the notion of the degree of compactness and prove a test for compactness.
Zentralblatt review: We investigate foliations on smooth manifolds that are determined by a closed 1-form with Morse singularities. The problem of investigating the topological structure of level surfaces for such a form was posed by S. P. Novikov [Russ. Math. Surv. 37, No. 5, 1-56 (1982); translation from Usp. Mat. Nauk 37, No. 5(227), 3-49 (1982; Zbl 0571.58011)]. This problem was treated in [S. P. Novikov, Tr. Mat. Inst. Steklova 166, 201-209 (1984; Zbl 0553.58005); A. V. Zorich, Math. USSR, Izv. 31, No. 3, 635-655 (1988); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 51, No. 6, 1322-1344 (1987; Zbl 0668.58004); Le Ty Kuok Tkhang, Math. Notes 44, No. 1/2, 556-562 (1988); translation from Mat. Zametki 44, No. 1, 124-133 (1988; Zbl 0656.58007); L. A. Alaniya, Russ. Math. Surv. 46, No. 3, 211-212 (1991); translation from Usp. Mat. Nauk 46, No. 3(279), 179-180 (1991; Zbl 0736.58001)]. The present paper is devoted to the compactness problem for level surfaces. We introduce the notion of degree of compactness and prove a test for compactness expressed in the terms of the degree. The present paper is a natural continuation of [the author, Math. Notes 53, No. 3, 356-358 (1993); translation from Mat. Zametki 53, No. 3, 158-160 (1993; Zbl 0809.57018)].
MSC: 57R30 (according to Zentralblatt)
JCR impact factor: 0.157 (2001), 0.337 (2009)
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Russian: info and full text on the journal's site (open access). The title above links to a locally mirrored file.
I.A. Mel'nikova. A test for non-compactness of the foliation of a Morse
form. Russ. Math. Surv. 50(2):444–445, 1995;
Zbl 0859.58005.
Translated from: И.А. Мельникова.
Признак некомпактности слоения морсовской формы.
УМН 50(3):217–218, 1995.
Erratum: The paper deals with (non-)compactifiable foliations and not (non-)compact.
The paper has no abstract.
MSC: 57R30, 53C12, 58Kxx
Zentralblatt review: The author studies foliations determined by a closed 1-form with Morse singularities on smooth compact manifolds. More precisely, the author investigates the problem of the existence of a non-compact leaf, verifies a test for non-compactness of a foliation in terms of the degree of irrationality of the considered 1-form, and shows that the non-compactness of a foliation is a case of general position.
Keywords: Morse form; foliations; non-compactness (according to Zentralblatt)
MSC: 58C25, 58K99, 58E05 (according to Zentralblatt)
JCR impact factor: 0.363 (2001), 0.425 (2009)
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I.A. Mel'nikova. An indicator of the noncompactness of a foliation on $M_g^2$.
Math. Notes 53(3):356–358, 1993;
Zbl 0809.57018.
Translated from: И.А. Мельникова.
Признак некомпактности слоения на
$M_g^2$.
Матем. заметки 53(3):158–160, 1993.
Erratum: The paper deals with (non-)compactifiable foliations and not (non-)compact.
The paper has no abstract.
Zentralblatt review: Let $\omega$ be a closed form on a manifold $M$ and possessing nondegenerate isolated singularities. A point $p \in M$ is called a regular singularity of $\omega$, if in some neighbourhood $O(p)\omega = df$, where $f$ is a Morse function having a singularity at $p$. The form $\omega$ determines a foliation $F_\omega$ on the set $M - Sing\omega$. Let $M = M^2_g$, the orientable closed surface of genus 2. The homology classes $[\gamma]$ of the nonsingular compact leaves of $F_\omega$ generate a subgroup of $H_1(M^2_g)$ denoted by $H_\omega$. If $[z_1], ..., [z_{2g}]$ is a basis of $H_1(M^2_g)$ we define $dirr\omega = rk_\mathbb{Q}{\int_{z_1}\omega, ..., \int_{z_{2g}}\omega} - 1$. By $M_\omega$ is denoted the set obtained by discarding all maximal neighbourhoods consisting of diffeomorphic compact leaves and all leaves which can be compactified by adding singular points. Theorem 1. $M_\omega = \emptyset \Leftrightarow rk H_\omega = g$. Theorem 2. If $\omega$ is a closed form with Morse singularities given on $M^2_g$ ($g \ne 0$) such that $dirr\omega \ge g$, then the foliation $F_\omega$ has a noncompact fiber.
JCR impact factor: 0.157 (2001), 0.337 (2009)
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Irina Gelbukh. Rank of a maximal subgroup in $H^1(M,Z)$ with trivial cup-product. Fourth International Conference of Applied Mathematics and Computing (FICAMC), Plovdiv, Bulgaria, August 12–18, 2007.
Abstract: Let $M$ be a smooth closed oriented manifold, $h^{max}(M)$ be the maximal rank of a maximal subgroup in $H^1(M,Z)$ with trivial cup-product, and $h^{min}(M)$ the minimal rank of such a subgroup. It has been shown that the value of $h(M)$ characterizes the topology of Morse form foliations on $M$: e.g., if $rk \omega > h(M)$, where $\omega$ is a Morse form on $M$, then its foliation has a minimal component. We give upper and lower bounds on $h{max}(M)$ and $h^{min}(M)$ in terms of the first and second Betti numbers. In addition, we calculate these values for a connected sum and direct product of manifolds.
Only abstract was published.
Proceedings; locally mirrored copy of the proceedings.
I.A. Melnikova. Compact foliations of a Morse form on $M_g^2$. International Conference on Topology and Applications in memory of P.S. Alexandroff (1896-1982). Moscow, Russia, May 27-31, 1996.
Erratum: The talk deals with compactifiable foliations and not compact.
This talk had no publication.
I.A. Mel'nikova. Compact foliations of Morse
forms. PhD thesis (in Russian). MSU, 1995.
И.А. Мельникова.
Компактные слоения морсовских форм. Дисс... к.ф.-м.н., МГУ, 1995.
Erratum: The thesis deals with compactifiable foliations and not compact.
Summary: Sufficient condition for compactifiability of a Morse form foliation, an upper bound on the rank of a Morse form defining a compactifiable foliation, and a lower bound on the number of the conic singularities of a Morse form defining a compactifiable foliation are given.
Abstracts in separate files, for Google Scholar's pleasure.
Last modified 2011-06-08