Geometric mechanics on Riemannian manifolds : applications to partial differential equations Differential geometry techniques have very useful and important applications in partial differential equations and quantum mechanics. This work presents a purely geometric treatment of problems in physics involving quantum harmonic oscillators quartic oscillators minimal surfaces Schrödingers Einsteins and Newtons equations and others. Historically problems in these areas were approached using the Fourier transform or path integrals although in some cases e.g. the case of quartic oscillators these methods do not work. New geometric methods which have the advantage of providing quantitative or at least qualitative descriptions of operators many of which cannot be treated by other methods are introduced. And conservation laws of the Euler Lagrange equations are employed to solve the equations of motion qualitatively when quantitative analysis is not possible. Main topics include: Lagrangian formalism on Riemannian manifolds; energy momentum tensor and conservation laws; Hamiltonian formalism; Hamilton Jacobi theory; harmonic functions maps and geodesics; fundamental solutions for heat operators with potential; and variational approach to mechanical curves. The text is enriched with good examples and exercises at the end of every chapter. Geometric Mechanics on Riemannian Manifolds is a fine text for a course or seminar in the above areas. It is also an ideal resource for mathematicians applied mathematicians graduate and advanced undergraduate students interested in elliptic and hyperbolic differential equations differential geometry calculus of variations quantum mechanics and physics. TOC:Introductory Chapter * Laplace Operator on Riemannian Manifolds * Lagrangian Formalism on Riemannian Manifolds * Harmonic Maps from a Lagrangian viewpoint * Conservation Theorems * Hamiltonian Formalism * Hamilton Jacobi Theory * Minimal Hypersurfaces * Radially Symmetric Spaces * Fundamental solutions for heat operators with potentials * Fundamental solutions for elliptic operators * Mechanical curves * Bibliography * Index