The convention is $\bar{\nabla} = \nabla + \sigma$ where $\bar{\nabla} $ is the connection on $\mathbb{R}^{n+1}$, $\nabla$ is the connection on $S$ and $\sigma$ is the second fundamental tensor. The laplacian sign convention I have already described. I dont think it is a problem of sign convention. Any other opinions ?

well you shd clarify what you mean by the sum of two ideals. if the ring is noncommutative, then the usual definition of sum of two ideals namely all linear combinations of the form $a_1x_1+a_2x_2$ for $x_1\in I$ and $x_2\in J$ does not make any sense ... for all you know the two ideals might be right ideals. the only way to make sense of sum of two ideals would be to just take all the elements of the form $x_1+x_2$ ... suppose you mean it then it is clear that $r(I)+r(J)\subset r(I\cap J)$ .. similarly for the left ideal case. so your question reduces to if $l(I\cap J) \subset l(I) + l(J)$ th