Faculty

Vladimir . DubovoyVladimir . Dubovoy Vladimir . Dubovoy Full professor of the department of pure mathematics, doctor of sciences in mathematics

Sergey  . FavorovSergey  . Favorov Sergey . Favorov Full professor of the department of pure mathematics, doctor of sciences in mathematics

Sergey . GefterSergey . Gefter Sergey . Gefter Associate professor of the department of pure mathematics, head of the department of pure mathematics, phd in mathematics

Vyacheslav . GordevskyyVyacheslav . Gordevskyy Vyacheslav . Gordevskyy Full professor of the department of pure mathematics, doctor of sciences in mathematics

Vladimir (Volodymyr) M. KadetsVladimir (Volodymyr) M. Kadets Vladimir (Volodymyr) M. Kadets Full professor of the department of pure mathematics, doctor of sciences in mathematics

Mariya . ShcherbinaMariya . Shcherbina Mariya . Shcherbina Professor of the department of pure mathematics

Dmitry . ShepelskyDmitry . Shepelsky Dmitry . Shepelsky Doctor of sciences in mathematics

Alexander L. YampolskyAlexander L. Yampolsky Alexander L. Yampolsky Professor of the department of pure mathematics, doctor of sciences in mathematics

Dmytry V. BolotovDmytry V. Bolotov Dmytry V. Bolotov Doctor of sciences in mathematics

Vasyl O. GorkavyyVasyl O. Gorkavyy Vasyl O. Gorkavyy Doctor of sciences in mathematics, associate professor

Alexander V. RezounenkoAlexander V. Rezounenko Alexander V. Rezounenko Professor of the department of pure mathematics, doctor of sciences in mathematics

Anna M. VishnyakovaAnna M. Vishnyakova Anna M. Vishnyakova Professor of the department of pure mathematics, doctor of sciences in mathematics

Tamara . FastovskaTamara . Fastovska Tamara . Fastovska Associate professor of the department of pure mathematics, phd in mathematics, associate professor

Nataliуa . GiryaNataliуa . Girya Nataliуa . Girya Associate professor of the department of pure mathematics, phd in mathematics

Oleksii . HukalovOleksii . Hukalov Oleksii . Hukalov Associate professor of the department of pure mathematics, phd in mathematics

Eugene . KarolinskyEugene . Karolinsky Eugene . Karolinsky Associate professor of the department of pure mathematics, phd in mathematics

Eugene V. PetrovEugene V. Petrov Eugene V. Petrov Phd in mathematics, senior lecturer

Aleksey . ShcherbinaAleksey . Shcherbina Aleksey . Shcherbina Phd in mathematics, senior lecturer

Olena O. ShugailoOlena O. Shugailo Olena O. Shugailo Phd in mathematics, senior lecturer

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Viktoria V. DavydovaViktoria V. Davydova Viktoria V. Davydova Engineer

Iryna V. KatsIryna V. Kats Iryna V. Kats Leading engineer

Thu Hien . NguyenThu Hien . Nguyen Thu Hien . Nguyen

Dmytro . SeliutinDmytro . Seliutin Dmytro . Seliutin

Olesia O. ZavarzinaOlesia O. Zavarzina Olesia O. Zavarzina Ph.d., associate professor

Alexander L. Yampolsky

Professor of the department of pure mathematics, doctor of sciences in mathematics

Link on external publications: scholar.google.com.ua.

List of selected publications

A. Yampolsky On stability of left invariant totally geodesic unit vector fields on three dimensional Lie group // Geometry and its Applications , Springer Proceedings in Mathematics & Statistics, Rovenski, Vladimir, Walczak, Paweł (Eds.) , 2014, Vol. 72 , p. 167 - 195, 2013

We consider the problem on stability or instability of unit vector fields on three-dimensional Lie groups with left-invariant metric which have totally geodesic image in the unit tangent bundle with the Sasaki metric with respect to classical variations of volume. We prove that among non-flat groups only SO(3) of constant curvature +1 admits stable totally geodesic submanifolds of this kind. Restricting the variations to left-invariant (i.e., equidistant) ones, we give a complete list of groups which admit stable/unstable unit vector fields with totally geodesic image.

Keywords: Sasaki metric, Lie group, stable submanifold

Yampolsky A. On geodesics of tangent bundle with fiberwise deformed Sasaki metric over Kahlerian manifold (2012) // Journal of Math. Phys., Analysis, Geom., 2012, v. 8/2, p. 177-189, 2012

We propose a fiber-wise deformation of the Sasaki metric on slashed and unit tangent bundles over the Kalerian manifold based on the Berger deformation of metric on a unit sphere. The geodesics of this metric have dierent projections on a base manifold for the slashed and unit tangent bundles in contrast to usual Sasaki metric. Nevertheless, the projections of geodesics of the unit tangent bundle over the locally symmetric Kahlerian manifold still preserve the property to have all geodesic curvatures constant.

Keywords: Sasaki metric, Kahlerian manifold, tangent bundle, geodesics.

Yampolsky A. Minimal and totally geodesic sections of the unit sphere bundles. // Visnyk KhNU, ser. Math. App. Math and Mech. , v. 1030, p. 54 – 70, 2012

We consider a real vector bundle $E$ of rank $p$ and a unit sphere bundle $E_1 \subset E$ over the Riemannian $M^n$ with the Sasaki-type metric. A unit section of $E_1$ gives rise to a submanifold in $E_1$. We give some examples of local minimal unit sections and present a complete description of local totally geodesic unit sections of $E_1$ in the simplest non-trivial case $p = 2$ and $n = 2$.

Yampolsky A. Totally geodesic vector fields on pseudo-Riemannian manifolds. // Visnyk Kharkiv Karazin Univ., ser Math, App. Math and Mech, , v. 990, p. 4 - 14, 2011

We consider the submanifolds in the unit tangent bundle of the pseudo- Riemannian manifold generated by the unit vector fields on the base. We have found the second fundamental form of this type of submanifolds with respect to the normal vector field of a special kind. We have derived the equations on totally geodesic non-isotropic unit vector field. We have found all the two-dimensional pseudo-Riemannian manifolds which admit non- isotropic totally geodesic unit vector fields as well as the fields.

Yampolsky A. Invariant totally geodesic unit vector fields on three-dimensional Lie groups // Journal of Mathematical Physics, Analysis, Geometry, vol. 3, No. 2, pp. 253 - 276, 2007

We give a complete list of left-invariant unit vector elds on three- dimensional Lie groups equipped with a left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group equipped with the Sasaki metric. As a result we obtain that each three-dimensional Lie group admits totally geodesic unit vector eld under some conditions on structural constants. From a geometrical viewpoint, the eld is either parallel or a characteristic vector eld of a natural almost contact structure on the group.

Keywords: Sasaki metric, totally geodesic unit vector eld, almost contact structure, Sasakian structure.

Yampolsky A. Totally geodesic unit vector fields on Riemannian manifold // Dokl. Ukr. Acad Nauk, v.3, p. 32-35, 2005

Totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold. // Journal of Mathematical Physics, Analysis, Geometry, v.1/1, p. 116-139, 2005

We give a full description of totally geodesic submanifolds in the tangent bundle of a Riemannian 2-manifold of constant curvature and present a new class of a cylinder-type totally geodesic submanifolds in the general case.

Yampolsky A. On special types of minimal and totally geodesic unit vector fields. // 7-th International Conference on Geometry, Integrability and Quantization, June 2-10, Varna (Bulgaria), SOFTEX, Sofia, p. 290 – 304, 2005

We present a new equation with respect to a unit vector field on Riemannian manifold $M^n$ such that its solution defines a totally geodesic submanifold in the unit tangent bundle with Sasaki metric and apply it to someclasses of unit vector fields. We introduce a class of covariantly normal unit vector fields and prove that within this class the Hopf vector field is a unique global one with totally geodesic property. For the wider class of geodesic unit vector fields on a sphere we give a new necessary and sufficient conditionto generatea totally geodesic submanifold in $T_1S^n$.

Keywords: Sasaki metric, minimal unit vector field, totally geodesic unit vector field, strongly normal unitvector field,Sasakian space form.

Abbassi M.T.K., Yampolsky A. Transverse totally geodesic submanifolds of the tangent bundle. // Math. Publ. Debrecen, v.64 /1-2, p. 129-154, 2004

It is well-known that if » is a smooth vector ¯eld on a given Rie- mannian manifold Mn then » naturally de¯nes a submanifold »(Mn) transverse to the ¯bers of the tangent bundle TMn with Sasaki metric. In this paper, we are interested in transverse totally geodesic subman- ifolds of the tangent bundle. We show that a transverse submanifold Nl of TMn (1 · l · n) can be realized locally as the image of a sub- manifold Fl of Mn under some vector ¯eld » de¯ned along Fl. For such images »(Fl), the conditions to be totally geodesic are presented. We show that these conditions are not so rigid as in the case of l = n, and we treat several special cases (» of constant length, » normal to Fl, Mn of constant curvature, Mn a Lie group and » a left invariant vector ¯eld).

Yampolsky A., Saharova E. Powers of the space form curvature operator and geodesics of the tangent bundle. // Ukr. Math. Journal, v.56/9, p. 1231-1243, 2004

A. Yampolsky Full description of totally geodesic unit vector field on Riemannian 2-manifold. // Matematicheskaya fizika, analiz, geometriya, 2004, v.11/3, p.355-365, 2004

We give a full geometrical description of local totally geodesic unit vector field on Riemannian 2-manifold, considering the field as a local embedding of the manifold into its unit tangent bundle with the Sasaki metric.

Keywords: Sasaki metric, totally geodesic unit vrctor field

A. Yampolsky Totally geodesic property of the Hopf vector field. // Acta Math. Hungarica, 2003, v.101, № 1-2, p. 73-92, 2003

We prove that the Hopf vector field is a unique one among geodesic unit vector fields on spheres such that the submanifold generated by the field is totally geodesic in the unit tangent bundle with Sasaki metric. As application, we give a new proof of stability (instability) of the Hopf vector field with respect to volume variation using standard approach from the theory of submanifolds and find exact boundaries for the sectional curvature of the Hopf vector field.

Keywords: Sasaki metric, Hopf vector field, curvature

A. Yampolsky On extrinsic geometry of unit normal vector field of Riemannian hyperfoliation. // Math. Publ. Debrecen, v.63/4, p. 555-567, 2003

We consider a unit normal vector feld of (local) hyperfoliation on a given Riemannian manifold as a submanifold in the unit tangent bundle with Sasaki metric. We give an explicit expression of the second fundamental form for this submanifold and a rather simple condition its totally geodesic property in the case of a totally umbilic hyperfoliation. A corresponding example shows the non-triviality of this condition. In the 2-dimensional case, we give a complete description of Riemannian manifolds admitting a geodesic unit vector feld with totally geodesic property.

Keywords: Sasaki metric, hyperfoliation

A. Yampolsky On the mean curvature of a unit vector field. // Math. Publ. Debrecen, v.60, 1/2, p. 131-155, 2002

We present an explicit formula for the mean curvature of a unit vector field on a Riemannian manifold, using a special but natural frame. As applications, we treat some known and new examples of minimal unit vector fields. We also give an example of a vector field of constant mean curvature on the Lobachevsky (n + 1) space.

Keywords: Sasaki metric, minimal unit vector field

A. Yampolsky On the intrinsic geometry of a unit vector field. // Comment. Math. Univ. Carolinae 2002,, v.43, № 2, p. 299-317, 2002

We study the geometrical properties of a unit vector field on a Riemann- ian 2-manifold, considering the field as a local imbedding of the manifold into its tangent sphere bundle with the Sasaki metric. For the case of con- stant curvature K, we give a description of the totally geodesic unit vector fields for K = 0 and K = 1 and prove a non-existence result for K= 0 and K=1. We also found a family of vector fields on the hyperbolic 2-plane L^2 of curvature -c^2 which generate foliations on T_1L^2 with leaves of constant intrinsic curvature -c^2 and of constant extrinsic curvature -c^2/4 .

Keywords: Sasaki metric, totally geodesic submanifold

Yampolsky A. On the vertical strong-sphericity index of Sasaki metric of tangent sphere bundle. // Math. phys., analysis and geometry, v.3, No ¾, p. 446-456, 1996

A distribution Lq on a Riemannian manifold is called strong spherical if the curvature tensor of its metric satisfy R(X,Y)Z=c( (Y,Z)X-(X,Z)Y ) (c =const > 0 ) for any X,Z TM and Y Lq The integer q = dim Lq is the index of sphericity. In the case of tangent sphere bundle one can regard some special strongly spherical distributions: vertical, horizontal and mixed. The most interesting case is vertical because the fibers are of constant curvature 1. The following assertions were proved: Theorem 1. The vertical sphericity index of Sasaki metric of T1M2 is 1 if and only if M2 is of constant curvature k and c=k2/4. Theorem 2. If Mn is locally symmetric and n 4 is even then the sphericity index of Sasaki metric  1

Yampolsky A. On the totally geodesic vector fields on submanifold // Math. phys., analysis and geometry, v.1, No 3/4, pp. 540-545, 1994

Let  be a smooth vector field defined on a submanifold Fk of Riemannian manifold Mn. This field generate naturally a submanifold (Fk) in the tangent bundle TMn . Endow TMn with Sasaki metric and (Fk) with induced one. The problem we solve is to find such a field that (Fk) is totally geodesic in TMn . We got: Theorem 1. If  is normal vector field on Fk parallel in normal connection then (Fk) is totally geodesic in TMn if and only if Fk is totally geodesic in Mn. Theorem 2. If  is normal vector field on Fk in a space of constant curvature Mn(k) then (Fk) is totally geodesic in TMn if and only if Fk is totally geodesic in Mn(k) and |  | = const.

Yampolsky A. On the strong sphericity index of Sasaki metric of tangent sphere bundle. // Ukr. geom. sbornik, v.35, pp. 150-159, 1992

A distribution Lq on a Riemannian manifold is called strong spherical if the curvature tensor of its metric satisfy R(X,Y)Z=c( (Y,Z)X-(X,Z)Y )(c=const > 0) for any X,Z TM and Y Lq The integer q = dim Lq is the index of sphericity. The following assertions were proved: Theorem1. Suppose that Sasaki metric of T1M2 is strongly spherical with sphericity value c. Then: a) q=1 if and only if M2 has constant Gaussian curvature k  1 and c =k2/4, b) q=3 if and only if M2 has constant Gaussian curvature k = 1, c) q=0 otherwise. Theorem 2. Suppose that Sasaki metric of T1Mn (n>2) is strongly spherical with sphericity value k. If c>1/3 and c  1 then q=0. Theorem 3. Let (Mn,k) be Riemannian manifold of constant curvature k. Suppose that Sasaki metric of T1(Mn,k) (n>2) is strongly spherical with sphericity value c. Then: a) q=1 if and only if k=1, c=1/4, b) q=0 otherwise.

Yampolsky A. On characterisation of projections of geodesics of tangent (sphere) bundle of complex protective space. // Ukr. geom. sbornik, v.34, pp. 121-126, 1991

It was known that projections of geodesics of tangent (sphere ) bundle of space forms are curves of constant first and second curvatures while others are zeroes. In this article it was proved that projections of geodesics of tangent (sphere) bundle of complex projective space are curves of constant curvatures k1, k2, k3, k4, k5 , while others are zeroes.

Yampolsky A., Borisenko A.A. Riemannian geometry of bundles. // Uspehi math. nauk, No 6, pp 51-95, 1991

This is our expository paper, which contains up-to-date state of art in the field of geometry of tangent and normal bundles with Sasaki-like metric and applications.

Yampolsky A., Borisenko A.A. Riemannian geometry of bundles (in Russian)(and in Eng) // Uspehi math. nauk, No 6, pp 51-95, 1991

This is our expository paper, which contains up-to-date state of art in the field of geometry of tangent and normal bundles with Sasaki-like metric and applications.(in Eng)

Yampolsky A. On the extremal values of sectional curvature of Sasaki metric of tangent sphere bundle of constant curvature space. // Ukr. geom. sbornik, v.32, pp. 127-137, 1989

Let (Mn,k) be Riemannian manifold of constant curvature k. The main result of this paper is the following: Theorem: The extremal values of sectional curvature K of T1 (Mn,k) are a) for n=2 Kmin= k (1-3/4 k) when k  (- ,0]  (1, +), Kmin = k2 /4 when k  (0,1], Kmax = k2 /4 when k  (- ,0]  (1, +), Kmax= k (1-3/4 k) when k  (0,1]; b) for n  3 Kmin= k (1-3/4 k) when k  (- ,0]  (4/3, +), Kmin= 0 when k  (0,1], Kmax =k+k2(k-5)2/ (4(k2-4k-1)) when k  (- ,(3-17)/2], Kmax = 1 when k  (3-17)/2, 2/3], Kmax = k+k2/ (4(2k-1)) when k  (2/3, (5+17)/2], Kmax = k2/4 when k  ((5+17)/2, +].

Yampolsky A.., Borisenko A.A. The Sectional curvature of Sasaki metric of the T1Mn. // Ukr. geom. sbornik, v.30 , pp.10-17, 1987

Here we studied the tangent bundle of vectors of fixed length  over general Riemannian manifold . We gave sufficient and closely necessary condition for the sectional curvature of the Sasaki metric on this bundle to be nonnegative in terms of value of  .

Yampolsky A., Borisenko A.A. On the Sasaki metric of tangent and normal bundle. // Dokl. Acad. Sci. USSR, v. 294, No 1, pp. 19-22, 1987

Here we announced the definition and basic properties of Sasaki-like metric on normal and normal sphere bundle of the submanifold in Riemannian space.

Yampolsky A., Borisenko A.A. On the Sasaki metric of normal bundle of submanifold in Riemannian space. // Math. Sbornik, v.134, No 2, pp. 158-176, 1987

This paper contains a detailed construction of Sasaki-like metric on the normal bundle of a submanifold in Riemannian space. The following analogies of "tangent" theorems were proved here: Theorem 1. The Sasaki metric of NFk is flat if and only if Fk is a manifold with intrinsically flat metric embedded in Mk+p with flat normal connection. Theorem 2. If the vertical nullity index of NFk with Sasaki metric is equal to q then on Fk there are q linearly independent normal vector fields parallel with respect to normal connection. For the case of spherical normal bundles, i.e. the normal bundle of vectors of constant length  , the Theorem 3 gives a sufficient condition for sectional curvature of NFk to be nonnegative.

Yampolsky A.., Borisenko A.A. Cylindricity of tangent bundles of strongly parabolic metrics and strongly parabolic surfaces. // Ukr. geom. sbornik, v.29, p. 12-32, 1986

Here we proved that if the intrinsic nullity of the Sasaki metric of a tangent bundle TMn is k, then k is even and Mn is the metric product of a Riemannian manifold Mn-k/2 by a Euclidean space Ek/2. As a consequence, TMn is the metric product of TMn-k/2 by Ek. An expression is obtained for the second fundamental forms of the imbedding TFk  TMn in terms of the second fundamental forms of the imbedding Fk  Mn and the curvature tensor of Mn. It was proved that TFk is totally geodesic in TMn if and only if Fk is totally geodesic in Mn.

Yampolsky A. On the curvature of Sasaki metric of tangent sphere bundle. // Ukr. geom. sbornik, v.28, p. 132-145, 1985

Here I study the sectional, Ricci and Scalar curvature curvatures of Sasaki metric on the tangent sphere bundle over the space of constant curvature k. The following assertions were proved: Theorem1. The sectional curvature of the Sasaki metric of the unit tangent sphere bundle of n-dimensional Riemannian of constant curvature k is non-negative if and only if 0k 4/3. Theorem2 . The sectional curvature K of the Sasaki metric of the unit tangent sphere bundle T1Sn of n-dimensional unit sphere Sn lies within the limits 0K 5/4. Also, the limits of variation of Ricci and Scalar curvature of Sasaki metric were found.

Yampolsky A. To the geometry of tangent sphere bundle of the Riemannian manifold. // Ukr. geom. sbornik, 1988, v.24, p.129-132, 1981

The tangent sphere bundle $T_rM^2$ of Riemannian manifold M of dimension 2 was considered. The main result is: The sectional curvature of Sasaki metric of $T_rM^2$ is positive if and only if $|grad K|^2 < K^3(1-3/4r^2 K)$.

Keywords: Sasaki metric, sectional curvature

Yampolsky, A Eikonal Hypersurfaces in the Euclidean n-Space. // Mediterranean Journal of Mathematics. 2017. 14: 160,

Yampolsky A., Fursenko O. Caustics of wave fronts reflected by a surface. // Mediterranean Journal of Mathematics. , 2017. 14: 160,

Yampolsky A., Opariy A. Generalized helices in three dimensional Lie groups // Turkish Journal of Mathematics., 2019. 43: 1447 – 1455. ,

Yampolsky A. Catacaustics of a hypersurface in the Euclidean n-space. // Mediterranean Journal of Mathematics., 2019. 16: 88,

Yampolsky A. On Projective Classication of Points of a Submanifold in the Euclidean Space // Journal of Mathematical Physics, Analysis, Geometry. , 2020. V. 16, № 3, P. 364–371.,