If $M$ is a spin manifold, then any submanifold of codimension 1 is also a spin manifold. This yields a lot of examples, for example, that $S^n$ is spin etc.

(I may not have understood your point completely.)

If $M$ is a spin manifold, then any submanifold of codimension 1 is also a spin manifold. This yields a lot of examples, for example, that $S^n$ is spin etc.

(I may not have understood your point completely.)

Edit: As pointed out in the comments, one has to demand as well that the submanifold is orientable.