A vector field induced de Rham-Hodge theory on manifolds

We introduce a de Rham-Hodge framework induced by a vector field on a compact, oriented smooth manifold. Using a vector field induced bundle isomorphism on differential forms, we define a vector field induced Hodge $L^2$-inner product, codifferential, and Hodge Laplacian. Unlike classical deformations, such as the drifting and Witten-type Hodge Laplacians, the induced Laplacian modifies the principal symbol and gives rise to an anisotropic Laplace-Beltrami type operator on functions. We establish the resulting de Rham-Hodge theory for closed manifolds, including the ellipticity of the induced Hodge Laplacian and the corresponding Hodge decomposition and isomorphism results. We further extend the framework to manifolds with boundary by imposing certain vector field induced boundary conditions, which are necessary to restore the adjointness between the differential and induced codifferential, and to obtain a well-posed boundary value problem. Under these boundary conditions, we establish analogues of the Hodge-Morrey and Friedrichs decompositions. We also discuss several structural properties of the framework, including its relation to anisotropic Laplace-Beltrami operators, its spectral behavior in several explicit examples, and its invariance under isometries.