  In real analysis, the Lojasiewicz inequalities, revitalized by Leon Simon in
his pioneering work on singularities of energy minimizing maps, have proven to
be monumental in differential geometry, geometric measure theory, and
variational problems. These inequalities provide specific growth and stability
conditions for prescribed real-analytic functions, and have found applications
to gradient flows, gradient systems, and as explicated in this paper, vector
bundles over compact Riemannian manifolds. In this work, we outline the theory
of functionals and variational problems over vector bundles, explore
applications to arbitrary real-analytic functionals, and describe the energy
functional on $S^{n-1}$ as a functional over a vector bundle.
