A Special Class of Tensor Product Surfaces with Harmonic Gauss Map
Azam Etemad Dehkordy*
Department of Mathematical sciences, Isfahan University of Technology, Isfahan, Iran
Abstract
In this paper, we stat a necessary and sufficient condition for Gauss map of the tensor product of planar unite circle and a special smooth curve in En to be harmonic. In this way, we construct two orthonormal basis for the tangent space and the normal space of the resulting tensor product surface. As a direct consequence of these basis, we also get a result about shape operators of this surface.
Keywords
Tensor Product, Gauss Map, Harmonic Map
Received: February 3, 2015
Accepted: April 9, 2015
Published online: April 17, 2015
@ 2015 The Authors. Published by American Institute of Science. This Open Access article is under the CC BY-NC license. http://creativecommons.org/licenses/by-nc/4.0/
1. Introduction
The tensor product of two immersions of a given Riemannian manifold was introduced by Chen in the late 1970,s ([3]). This notion is a generalization of the quadratic representation of a submanifold.
In a special case, a tensor product surface is obtained by taking the tensor product of two curves. A number of properties such as minimality and totally reality are studied about tensor product of two planar curves ([7]). Moreover, minimal and pseudo- minimal tensor product of Lorentzian planar curve and an Euclidean planar curve is considered by Mihai ([8]).
Gauss map is one of the topics in differential geometry. On the other hand, harmonic functions have very useful properties in advanced mathematics. So, we study the tensor product surfaces of two curves that have harmonic Gauss map.
2. Preliminaries
In this section, we recall some standard definitions and results from Riemannian geometry. Let M be an n-dimensional manifold, Em be an m-dimensional Euclidean space and be an isometric immersion, as well as
the Levi-Civita connection of
and
the induced connection on
from
We denote the second fundamental form of
in
by
normal connection in the normal bundle of
by
and the shape operator in the direction of normal vector field n by
. It is well known that the two later notions are related to each other by
< II(X, Y ),n >=<,Y > (1)
where and
are tangent vector fields to
For an n-dimensional submanifold
in
the mean curvature vector
is given by
If then the submanifold is said to be minimal. A submanifold is called totally geodesic if
Furthermore, the Gaussian and Weingarten formula are given, respectively, by
(2)
(3)
Using above notations, we have the following Ricci equation,
1 (4)
for tangent vector fields and normal vector fields
and
Let be the Grassmannian consisting of all oriented n-planes through the origin of
For an isometric immersion
the Gauss map
of
is a smooth map which carries
into the oriented n-plane in
, which obtained from the parallel translation of
, the tangent space of
at
in
. We known that
canonically imbedded in
, the vector space obtained by the exterior product of
vectors in
. We can assume
as Euclidean space
where
, so the Gauss map at
can be written as
)(p), where
is an adapted local orthonormal frame field in
such that
are tangent to
and
are normal to
If
be the set of real smooth functions on
, then the Laplacian of
is defined by
(5)
Note that in this context, smooth can be replaced by
3. A Special Tensor Product Surface in
Let be the unit planar circle centered at the origin with parameterization
) and
be a unit speed smooth curve in
with parametrization
Here, we consider
for every
(index 1 can be replaced by
The tensor product surface
of two curves
and
is given by,
Assume that defines an isometric immersion of
into
Let prime denote derivative with respect to
It is easily seen that
and
form an orthonormal frame for tangent space of . Moreover, an orthonormal basis normal to
is given by
where
and
.
If we use the following abbreviation,
and for
also for we us
we get,
(6)
and for we have,
(7)
where and
’s are in
and
One immediate result that follows from (3) and (7) is following Corollary.
Corollary. Let be the tensor product surface of the circle
) and unit speed smooth curve
, then for
, , we have
(8)
The following Theorem provides us a necessary and sufficient condition for our special tensor product surface to have harmonic Gauss map
Theorem. Let be a tensor product surface of the circle
) and unit speed smooth curve
of plane.
Proof. If we use (5), (6) and (7), then a direct computation shows that the Laplacian of the Gauss map is given by
(9)
where for
and
.
In definitions of and
, prime means the derivation respect to
. If the Gauss map of
is harmonic, i.e.
, then (9) implies that
(10)
, ...,
,
,
Since all terms on the right-hand side of the first equation in (10) are nonnegative, hence we have
for This result and (8), show that
is a totally geodesic surface in
and so
is a part of a plane. The converse is obvious.
4. Conclusion
The main conclusion of this paper is a planar surface in even dimensional Euclidean space can be obtained from tensor product of unit circle with a unit speed curve in an Euclidean surface of half dimension.
References