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
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).
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.
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 NFk 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 0k 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 0K 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.,
