Notes in Mathematics 1824

The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Preface Differential geometry is traditionally regarded as the study of smooth manifolds, but sometimes this framework is too restrictive since it does not admit certain basic geometric intuitions. On the contrary, these geometric constructions are possible in the broader category of differentiable spaces. Let us indicate some natural objects which are differentiable spaces and not manifolds: – Singular quadrics. Elementary surfaces of classical geometry such as a quadratic cone or a " doubly counted " plane are not smooth manifolds. Nevertheless, they have a natural differentiable structure, which is defined by means of the consideration of an appropriate algebra of differentiable functions. For example, let us consider the quadratic cone X of equation z 2 −x 2 −y 2 = 0 in R 3. It is a differentiable space whose algebra of differentiable functions is defined by A := C ∞ (R 3)/p X = C ∞ (R 3)/(z 2 − x 2 − y 2) , where p X stands for the ideal of C ∞ (R 3) of all differentiable functions vanishing on X. In other words, differentiable functions on X are just restrictions of differentiable functions on R 3. Let us consider a more subtle example. Let Y be the plane in R 3 of equation z = 0. Of course this plane is a smooth submanifold. On the contrary, the " doubly counted " plane z 2 = 0 makes no sense in the language of smooth manifolds. It is another differentiable space with the same underlying topological space (the plane Y) but a different algebra of differentiable functions: A := C ∞ (R 3)/(z 2). Note that A is not a subalgebra of C(Y, R), so that elements of A are not functions on Y in the set-theoretic sense. More generally, any closed ideal a of the Fréchet algebra C ∞ (R n) defines a differentiable space (X, A), where X := {x ∈ R n : f (x) = 0 for any f ∈ a} is the underlying topological space and A := C ∞ (R n)/a is the algebra of differentiable functions on this differentiable space. The pair (X, A) …