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

Example 3.38 Let M = C ∞ ([−1, 1] → lR); since M is a vector space, then it istrivially a manifold, with a single identity chart; and T c M = M. Let E : M → lRbe the evaluation functional E(f) = f(0), then DE(f; h) = h(0); let〈f, g〉 =∫ 1−1f(t)g(t) dt ;then the gradient of E would be δ 0 (the Dirac’s delta), that is a distribution (ormore simply a probability measure) but not an element of M.3.8 Group actionsDefinition 3.39 Let G be a group, M a set. We say that G acts on M if thereis a mapG × M → Mg, m ↦→ g · mthat respects the group operations:• if e is the identity element then e · m = m, and• for any g, h ∈ G, m ∈ Mh · (g · m) = (h · g) · m . (3.5)If G is topological group, that is, a group with a topology such that the groupoperation is continuous, and M has a topology as well, we will require that theaction be continuous. Similarly for smooth actions between smooth manifolds.The action of a group G on M induces an equivalence relation ∼, definedbym 1 ∼ m 2 iff there exists g such that m 1 = g · m 2 .We can then define the following objects.Definition 3.40m.• The orbit [m] of m is the family of all m 1 equivalent to• The quotient M/G is the space M/ ∼ of equivalence classes.• There is a projection π : M → M/G, sending each m to its orbit π(m) =[m].We will see many examples of actions in Example 4.8.32