Metrics of curves in shape optimization and analysis - Andrea Carlo ...

Since φ k are homeomorphisms, then the topology of M is “identical” to thetopology of U. The rôle of the charts is to define the differentiable structure ofM mimicking that of U. See for example the definition of directional derivativein eqn. (3.4).3.3.1 SubmanifoldDefinition 3.19 Suppose A, B are open subsets of two linear spaces. A diffeomorphismis an invertible C 1 function φ : A → B, whose inverse φ −1 is againC 1 .Let U be a fixed closed linear subspace of a l.c.t.v.s. X.Definition 3.20 A submanifold is a subset Mof X, such that, at any point c ∈ M we may finda chart φ k : U k → V k , with V k , U k ⊂ X open sets,c ∈ V k . The maps φ k are diffeomorphisms, andφ k maps U ∩ U k onto M ∩ V k .Most often, M = {Φ(c) = 0} where Φ : X → Y ; so, to prove that M is asubmanifold, we will use the implicit function theorem.Note that M is itself an abstract manifold, and the model space is U; sodim(M) = dim(U) ≤ dim(X). Vice versa any abstract manifold is a submanifoldof some large X (by well known embedding theorems).3.4 Tangent space and tangent bundleLet us fix a differentiable manifold M, and c ∈ M. We want to define thetangent space T c M of M at c.Definition 3.21 • The tangent space T c M is more easily described forsubmanifolds; in this case, we choose a chart φ k and a point x ∈ U k s.t.φ k (x) = c; T c M is the image of the linear space U under the derivativeD x φ k . T c M is itself a linear subspace in X. In figure 5 on the followingpage, we graphically represent T c M though as an affine subspace, bytranslating it to the point c.• The tangent bundle T M is the collection of all tangent spaces. If M isa submanifold of the vector space X, thenT M := {(c, h) | c ∈ M, h ∈ T c M} ⊂ X × X .The tangent bundle T M is itself a differentiable manifold; its charts are of theform (φ k , Dφ k ), where φ k are the charts for M.XU 1Mφ 1UV 1c26