  Let $I$ denote an ideal of a local Gorenstein ring $(R, \mathfrak m)$. Then
we show that the local cohomology module $H^c_I(R), c = \height I,$ is
indecomposable if and only if $V(I_d)$ is connected in codimension one. Here
$I_d$ denotes the intersection of the highest dimensional primary components of
$I.$ This is a partial extension of a result shown by Hochster and Huneke in
the case $I$ the maximal ideal. Moreover there is an analysis of connectedness
properties in relation to various aspects of local cohomology. Among others we
show that the endomorphism ring of $H^c_I(R)$ is a local Noetherian ring if
$\dim R/I = 1.$
