By Constantin Udrişte (auth.)
The item of this ebook is to give the fundamental proof of convex services, commonplace dynamical platforms, descent numerical algorithms and a few laptop courses on Riemannian manifolds in a kind appropriate for utilized mathematicians, scientists and engineers. It includes mathematical details on those matters and purposes disbursed in seven chapters whose subject matters are just about my very own components of analysis: Metric houses of Riemannian manifolds, First and moment adaptations of the p-energy of a curve; Convex services on Riemannian manifolds; Geometric examples of convex capabilities; Flows, convexity and energies; Semidefinite Hessians and functions; Minimization of capabilities on Riemannian manifolds. the entire numerical algorithms, computing device courses and the appendices (Riemannian convexity of services f:R ~ R, Descent equipment at the Poincare aircraft, Descent tools at the sphere, Completeness and convexity on Finsler manifolds) represent an try and make accesible to all clients of this ebook a few simple computational recommendations and implementation of geometric constructions. To extra relief the readers,this e-book additionally encompasses a a part of the folklore approximately Riemannian geometry, convex capabilities and dynamical platforms since it is sadly "nowhere" to be present in an analogous context; present textbooks on convex capabilities on Euclidean areas or on dynamical platforms don't point out what occurs in Riemannian geometry, whereas the papers facing Riemannian manifolds frequently steer clear of discussing basic proof. frequently a convex functionality on a Riemannian manifold is a true valued functionality whose limit to each geodesic arc is convex.
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Additional resources for Convex Functions and Optimization Methods on Riemannian Manifolds
P vanish for t = 0 has < n, we observe that the = ~~, satisfies J(O) = 0, J(1) '" O. Example. Suppose that the curvature tensor is identically zero on M. Then on M there exist n linearly independent parallel vector fields PI' i = 1, ... ,no Setting J(t) n = L fl(t) P (t), i=l I the Jacobi 1 1 =at+b, i reduces to So a Jacobi field along equation = 1, ... ,no r can have at most one zero. Thus there are no conjugate points, and E ** is nondegenerate. p Let « : (- c,c) x [0,1] ~ be a C~ map such that each oc(u)(t) = r(t).
Aex ] . II at au at 1 at (u ,u ) Now, all we have to do is to evaluate 1 (0,0) . 2 For this we take into account succesively aex = 0, at ~t 'J au 1 'J V = at 'atJ aex = 0 at aex au 1 (~ 'J = at w1 = -ex(O,O) is a geodesic), ' This completes the proof. The second variation formula and show following consequence is true. 3. 2. Cora llary . First and second variations of the p-energy of a curve §4. NULL 41 SPACE OF THE HESSIAN OF THE P-ENERGY = dr dt V Let r be a geodesic and be the tangent vector field of r.
2) Let (M,g) be a complete Riemannian manifold. If f: M ~ of class C1 and V f '" D, then the level sets of f ~ is are complete hypersurfaces of M. Applications to Hamiltonian systems Let (M, g .. ) be an n-dimensional Riemannian manifold with local coordinates IJ (Xi). Metric properties of Riemannian manifolds 27 (matrix notations, ). We observe that G-1 =[ The natural almost complex structure of TM is [~ J 1 r Let V: M ~ ~ be a COO potential on M and the Hamiltonian H: TM --7 ~, H=E E: TM ~ ~, E(x, y) = -2 g + V, where 1 I i j (x) y y j is the kinetic energy attached to the Riemannian structure of M.
Convex Functions and Optimization Methods on Riemannian Manifolds by Constantin Udrişte (auth.)