Let $M_0$ be a Riemannian manifold, $M_1$ a geodesic sphere of $M_0$ and $M_2$ an isometrically immersed submanifold of $M_1$, ie: $$ M_2 \subset M_1 \subset M_0$$
Take $X \in M_2$, and:
- $T_2$ the tangent space of $M_2$ at $X$,
- $N_{2,1}$ the normal space of $M_2$ at $X$, in the tangent space of $M_1$ at $X$,
- $N_{2,0}$ the normal space of $M_2$ at $X$,
- $N_{1,0}$ the normal space of $M_1$.
We can write: $$T_X M_0=T_2\oplus N_{2,0}=T_2 \oplus N_{2,1} \oplus N_{1,0}$$
Let:
- $H_{1,0}$ the mean curvature vector of the geodesic sphere $M_1$ in $M_0$: $H_{1,0}(u,v)\in N_{1,0}$
- $H_{2,1}$ the mean curvature vector of $M_2$ in $M_1$: $H_{2,1}(u,v)\in N_{2,1}$
- $H_{2,0}$ the mean curvature vector of $M_2$ in $M_0$: $H_{2,0}(u,v)\in N_{2,0}$
for $u,v \in T_2$.
Would you know under which conditions the following holds: $H_{2,0}=H_{2,1} + H_{1,0}$?
Many thanks in advance!