Deformations of a Courant Algebroid (E; < , >, \rho,o) and its Dirac subbundle have been widely considered under the assumption that the pseudo-Euclidean metric < , > fixed. We want to attack the same problem in a setting that allows < , > to deform.
By Roytenberg, a Courant Algebroid is equivalent to a Symplectic graded Q-manifold of degree 2. From this viewpoint, we first extend the denition of a Q-vector eld X on a graded manifold M so that it also encodes the compatible geometric structures such as a Symplectic structure, and then define the submaniold M M of "coisotropic type" which naturally generalizes the concept of Dirac subbundles. It turns out the simultaneous deformations of X and M can be controlled by an L1-algebra under certain regularity conditions of X. This result applies to the deformations of a Courant algebroid and itsDirac structures, the deformations of a Poisson manifold and its coisotropic submanifold,the deformations of a Lie algebroid and its Lie subalgebroid, and hopefully more.