1. Introduction
Let
M be an
-dimensional complete and simply connected Riemannian manifold with metric
g. Given a gradient vector field
for a smooth function
, if a one-parameter family of diffeomorphisms generated by the integral curves of
on
M satisfies
for the scalar curvature
R and some constant
, then
M is called a gradient steady, expanding and shrinking Yamabe soliton for
,
and
, respectively. In [
1], the warped product structure of a gradient Yamabe soliton is shown. But is well-known that the equation
for smooth functions
determines the warped product structure
where
and
is the warping function [
2]. Thus this fact can be applied to a gradient Yamabe soliton [
1] and a Ricci soliton with a concircular potential field [
3].
In another way by using the Jacobi differential equation, we show that the Equation (
2) determines the warped product structure. Note that each fiber of a warped product is called totally umblic if the shape operator is a multiple of the identity at each point. A gradient Yamabe soliton with
is a warped product with the totally umblic fibers
by the following shape operator
(
4).
Let
be a regular hypersurface of
M with a unit normal vector field
for some
. For orthogonal distributions
and
on
, we see that
for all
. Put
. Since
for all
, we get
So we can consider each level hypersurface
of
along a unit-speed geodesic
orthogonal to
with
and
by (
3). The shape operator of each level hypersurface
of
along a geodesic
is given by
for all
. The trace of the shape operator is the mean curvature
Since for all , is constant on each level hypersurface.
Differentiation of the Equation (
2)
with the Einstein convention gives
which implies
Take the trace in
k and
j, then we have
The trace of (
2)
gives
So we obtain
for all
as in [
4]. Here we show that the mean curvature
(
5) and the Equation (
6) determine the warped product structure. Since
Hence we obtain the Raychaudhuri Equation (
12) along a geodesic
with the zero shear tensor
Thus we can show Theorem 1 by the Jacobi differential equation.
Theorem 1. Let M be a complete and simply connected Riemannian manifold. If for smooth functions , then M is a warped product where and is the warping function.
For a gradient Yamabe soliton, we get
Thus a gradient Yamabe soliton with a regular hypersurface
for some
is a warped product with the totally umblic fibers whose scalar curvature are constant by (
10) (cf. [
1]). Differentiation of the above equation
for
shows that if
or
, then we get
. Thus under the assumption
or
, if a singular point is allowed, then
for some
. Otherwise
M is a product manifold.
A vector field
v on
M is said to be concircular if it satisfies
for all
and a non-trivial function
on
M. The warped product structure of a Ricci soliton with a concircular potential field is shown in [
3]. It is also pointed out that the gradient
of
f is a concircular vector field if and only if
. Then a gradient Ricci soliton equations
with
becomes
So
M is Einstein. Thus
is constant under
. In Theorem 5.1 in [
3],
M is turned out to be Ricci-flat. So we have
. Therefore we have
If a singular point is allowed, then for some . A Gaussian gradient Ricci soliton with on is an example for it.
We show the warped product structure with the base B whose dimension is .
Theorem 2. Let M be a complete and simply connected Riemannian manifold with . Assume that and are orthogonal distributions on andsuch that are unit normal vectors of for smooth functions and . If and , then M is a warped product with the warping function . 2. The Raychaudhuri and Jacobi Equation
Let
M be an
-dimensional Riemannian manifold and
be a regular hypersurface of
M with a unit normal vector field
for some
. Consider each level hypersurface
of
along a unit-speed geodesic
orthogonal to
with
and
. The Riemannian curvature tensor of
M along
, the shape operator of
is denoted by
,
, respectively. If a smooth
tensor field
satisfies
with initial conditions
and
for the identity endomorphism
of
, then
A is said to be an
H-Jacobi tensor along
.
Put
and
. Then it follows from [
5] that
and the shape operator of each level hypersurface
is
Thus we get the mean curvature
of
. For the adjoint ∗, the vorticity
becomes zero, since a variation tensor field
A is a Lagrange tensor (Proposition 1 in [
5]). The trace of (
11) gives the
Raychaudhuri equation
for the shear tensor
and the Ricc tensor
. The expansion
satisfies
as in [
5]. Differentiation of
gives
By (
12) and (
13), we have the Jacobi differential equation