For a compact Riemannian
m-manifold
endowed with a nontrivial conformal vector field
with a conformal factor
, there is an associated skew-symmetric tensor
called the associated tensor, and also,
admits the Hodge decomposition
, where
satisfies
, which is called the Hodge vector, and
is the Hodge potential of
. The main purpose of this article is to initiate a study on the impact of the Hodge vector and its potential on
. The first result of this article states that a compact Riemannian
m-manifold
is an
m-sphere
if and only if (1) for a nonzero constant
c, the function
is a solution of the Poisson equation
, and (2) the Ricci curvature satisfies
. The second result states that if
has constant scalar curvature
, then it is an
if and only if the Ricci curvature satisfies
and the Hodge potential
satisfies a certain static perfect fluid equation. The third result provides another new characterization of
using the affinity tensor of the Hodge vector
of a conformal vector field
on a compact Riemannian manifold
with positive Ricci curvature. The last result states that a complete, connected Riemannian manifold
,
, is a Euclidean
m-space if and only if it admits a nontrivial conformal vector field
whose affinity tensor vanishes identically and
annihilates its associated tensor
.
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