The scalar product on $\langle\,\cdot\,,\,\cdot\,\rangle$ of signature $(p+1, q+1)$ on $V$ induces the conformal structure on the projectivized null-cone and in fact this projectivised null-cone is a (flat) model space for the conformal geometry of signature $(p,q)$ (in the sense of Cartan geometry). In fact, the orthogonal group of your scalar product acts by conformal mappings on this projectivised null-cone and a stabilizier of it's point is a prabolic subgroup.

If you look more closely at the case of signature $(1,n)$ you'll be able to see that the projectivized null-cone is diffeomorphic to a sphere and it's not hard to work out which elements of $SO(p+1, q+1)$ act by which conformal transformations. The details for general signatures are in Slovák's lecture notes that I have linked to in a previous answer of mine.

As for your main question, I do not know the answer on top of my head, but maybe you can check out Conformal submanifold geometry I-III by Burstall, Calderbank.

The scalar product on $\langle\,\cdot\,,\,\cdot\,\rangle$ of signature $(p+1, q+1)$ on $V$ induces the conformal structure on the projectivized null-cone and in fact this projectivised null-cone is a (flat) model space for the conformal geometry of signature $(p,q)$ (in the sense of Cartan geometry). In fact, the orthogonal group of your scalar product acts by conformal mappings on this projectivised null-cone and a stabilizier of it's point is a prabolic subgroup.

If you look more closely at the case of signature $(1,n)$ you'll be able to see that the projectivized null-cone is diffeomorphic to a sphere and it's not hard to work out which elements of $SO(p+1, q+1)$ act by which conformal transformations. The details for general signatures are in Slovák's lecture notes that I have linked to in a previous answer of mine.

As for your main question, I do not know the answer on top of my head, but maybe you can check out Conformal submanifold geometry I-III by Burstall, Calderbank.

The scalar product on $\langle\,\cdot\,,\,\cdot\,\rangle$ on $V$ induces the standard conformal structure on the projectivized null-cone. If you look more closely at the case of signature $(1,n)$ you'll be able to see that the projectivized null-cone is diffeomorphic to a sphere. The details for general signatures are in Slovák's note that I have linked to in a previous answer of mine which is quite related to your problem.

As for your main question, I do not know the answer on top of my head, but maybe you can check out Conformal submanifold geometry I-III by Burstall, Calderbank.

The scalar product on $\langle\,\cdot\,,\,\cdot\,\rangle$ of signature $(p+1, q+1)$ on $V$ induces the conformal structure on the projectivized null-cone and in fact this projectivised null-cone is a (flat) model space for the conformal geometry of signature $(p,q)$ (in the sense of Cartan geometry). In fact, the orthogonal group of your scalar product acts by conformal mappings on this projectivised null-cone and a stabilizier of it's point is a prabolic subgroup.

If you look more closely at the case of signature $(1,n)$ you'll be able to see that the projectivized null-cone is diffeomorphic to a sphere and it's not hard to work out which elements of $SO(p+1, q+1)$ act by which conformal transformations. The details for general signatures are in Slovák's lecture notes that I have linked to in a previous answer of mine.

As for your main question, I do not know the answer on top of my head, but maybe you can check out Conformal submanifold geometry I-III by Burstall, Calderbank.