Moving frame and spin field representations of submanifolds in flat space

We introduce a spin field approach, that is compatible with the Cartan moving frame method, to describe the submanifold in a flat space. In fact, we consider a kind of spin field, that satisfies a Killing spin field equation (analogous to a Killing spinor equation) written in terms of the Clifford algebra, and we use the spin field to locally rotate the orthonormal basis \eI\. Then, the deformed orthonormal frame \I\ can be seen as the moving frame of a submanifold. We find some solutions to the Killing spin field equation and demonstrate an explicit example. Using the product of the spin fields, one can easily generate a new immersion submanifold, and this technique should be useful for studies in geometry and physics. Through the spin field, we find a linear relation between the connection and the extrinsic curvature of the submanifold. We propose a conjecture that any solution of the Killing spin field equation can be locally written as the product of the solutions we find.