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. //
Journal of Math. Phys., Analysis, Geom.,
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 $n = 2$.
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.
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.
//
Dokl. Ukr. Acad Nauk,
v.3, p. 32-35, 2005
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.
//
Math. Publ. Debrecen,
v.64 /1-2, p. 129-154, 2004
Yampolsky A., Saharova E.
//
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.
//
Math. phys., analysis and geometry,
v.3, No ¾, p. 446-456, 1996
Yampolsky A.
//
Math. phys., analysis and geometry,
v.1, No 3/4, pp. 540-545, 1994
Yampolsky A.
//
Ukr. geom. sbornik,
v.35, pp. 150-159, 1992
Yampolsky A.
//
Ukr. geom. sbornik,
v.34, pp. 121-126, 1991
Yampolsky A., Borisenko A.A.
//
Uspehi math. nauk,
No 6, pp 51-95, 1991
Yampolsky A.
//
Ukr. geom. sbornik,
v.32, pp. 127-137, 1989
Yampolsky A.., Borisenko A.A.
//
Ukr. geom. sbornik,
v.30 , pp.10-17, 1987
Yampolsky A., Borisenko A.A.
//
Dokl. Acad. Sci. USSR,
v. 294, No 1, pp. 19-22, 1987
Yampolsky A., Borisenko A.A.
//
Math. Sbornik,
v.134, No 2, pp. 158-176, 1987
Yampolsky A.., Borisenko A.A.
//
Ukr. geom. sbornik,
v.29, p. 12-32, 1986
Yampolsky A.
//
Ukr. geom. sbornik,
v.28, p. 132-145, 1985
Yampolsky A.
To the geometry of tangent sphere bundle of the Riemannian manifold. //
Ukr. geom. sbornik,
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
A. Yampolsky A, O.Fursenko
Caustics of wave fronts reflected by a surface //
Journal of Mathematical Sciences and Modeling,
2018, V 1 , Issue 2, Pages 131 - 137. https://doi.org/10.33187/jmsm.431543,
A. Yampolsky, A. Opariy
Generalized helices in three-dimensional Lie groups //
, Turkish Journal of Mathematics (2019) ,
43: 1447 – 1455. http://journals.tubitak.gov.tr/math/,
A. Yampolsky
Catacaustics of a hypersurface in the Euclidean n-space. //
Mediterranean Journal of Mathematics, ,
(2019) 16: 88. https://doi.org/10.1007/s00009-019-1365-3,
A. Yampolsky
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. ,
A. Yampolsky
On properties of the Reeb vector field of (α,β) trans-Sasakian structure. //
Turkish Journal of Mathematics, 2022,,
Vol. 46: No. 6, Article 19. ,
