Basic notions of Differential Geometry Md. Showkat Ali 7th November 2002 Manifold Manifold is a multi dimensional space. Riemann proposed the concept of ”multiply extended quantities” which was refined for the next fifty years and is given today by the notion of a manifold. Definition. A Hausdorff topological space M is called an n-dimensional (topological) manifold (denoted by M n) if every point in M has an open neighbourhood which is homeomorphic to an open set in Rn. Example 1. A surface R2 is a two-dimensional manifold, since φ : R2 R2 is homeomorphic. −→ Charts, Ck-class and Atlas Definition. A pair (U, φ) of a topological manifold M is an open subset U of M called the domain of the chart together with a homeomorphism φ : U V of U onto an open set V in Rn. Roughly−→ speaking, a chart of M is an open subset U in M with each point in U labelled by n-numbers. Definition. A map f : U Rm is said to be of class Ck if every func- tion f i(x1, ..., xn), i = 1, ...,−→ m is differentiable upto k-th order. The map m f : U R is called smooth if it is of C∞-class. −→ Definition. An atlas of class Ck on a topological manifold M is a set (Uα, φα), α I of each charts, such that { ∈ } (i) the domain Uα covers M, i.e. M = αUα, ∪ (ii) the homeomorphism φα satisfy the following compativility condition : 1 the map 1 φα φ− : φβ(Uα Uβ) φα(Uα Uβ) ◦ β ∩ −→ ∩ must be of class Ck. 1 The map φα φ− can be represented by a set of n-functions of n-variables, ◦ β x1 = f 1(y1, ..., yn), ... ... ... ... ... ... xn = f n(y1, ..., yn), that is, all f i, i = 1, ..., n can be differentiable upto k-th order. If k = , the above definition, the atlas is called smooth atlas. Two Ck at- ∞ lases (Uα, φα) and (Uβ, φβ) are called equivalent if their sum (Uα,Uβ), (φα, φβ) is an{ atlas of C}k. { } { } A topological manifold M, together with an equivalence class of Ck at- lases, is called a Ck structure on M and M is called Ck manifold and if k = , then M is called smooth manifold. ∞ Smooth manifold Definition. A smooth manifold M of dimension n is a Hausdorff topo- logical space for which i) each point p M has a neighbourhood Up which is homeomorphic to an open subset of R∈n. 1 ii) If Up Uq = φ then φp φq− : φq(Up Uq) φp(Up Uq) is a smooth diffeomorphism.∩ 6 (Smooth means◦ : all partial∩ derivatives→ of∩ all orders exist and are continuous.) Example 1. Let Sn denote the unit sphere in Rn+1, i.e. Sn = x Rn+1 x2 + ... + x2 = 1 { ∈ | 1 n+1 } is a smooth manifold of dimension n. 2 Example 2. Let Pn denote the set of all lines through the origin in Rn+1 is called the n dimensional real projective space denoted by RPn. So, RPn is a smooth manifold. Functions on Smooth manifolds A function f : M R on smooth (C∞) manifold M is called smooth at a point x M if in→ a coordinate chart (U, φ) with x U. The function 1 ∈ ∈ f φ− is smooth in the usual sense. (i.e. Smooth as a function of n-variables x1◦, ..., xn ; n = dim M.) 1 1 Remark 1. f φ− is just a usual function of n-variables, f φ− = fˆ(x1, ..., xn). ◦ ◦ Remark 2. The notion of smoothness of f : M R at x M does not depend on the choice of a chart. (If it is smooth at one→ particular∈ chart (U, φ) it is surely smooth at all others.) Definition. If a function f : M R is smooth at every point x M, then it is called a smooth function on→ the smooth manifold M. ∈ Differentiable Manifold A differentiable manifold is a topological space equipped with a collec- tion of charts. Each chart provides a one-to-one correspondence between open subsets of the manifold and open sets of the model space Rn, and charts with overlapping domains provide consistency in differentiability properties. 3 Definition. An n-dimensional differentiable manifold M is a Hausdorff topological space which satisfies the following conditions : i) for each Ui M there is a homeomorphism φi : Ui Vi, where Vi is an open set in Euclidean⊂ space Rn. → ii) If Ui Uj = φ, the homeomorphisms φi and φj combine to give a ∩ 6 1 homeomorphism φji = φj φi− of φi(Ui Uj) onto φj(Ui Uj) which is dif- ferentiable map. ◦ ∩ ∩ Example. The sphere S2 = (x, y, z) x2 + y2 + z2 = 1 is a differentiable manifold. { | } Complex Manifold Definition. A complex manifold is a smooth manifold locally modelled Cn 1 on the complex Euclidean space, and whose transition function φp φq− of open sets of Cn into Cn are holomorphic (complex analytic). ◦ Example Let Pn denote the set of all lines through the origin in Cn+1 is called the n dimensional complex projective space denoted by CPn. So, CPn is a complex manifold of real dimension 2n. Submanifold Definition. Let M be a differentiable manifold of dimension m and let N be a subset of M satisfying the conditions : i) N is a differentiable manifold of dimension n. ii) If p is a point of N, there is a local coordinate neighbourhood U of p in M with a local coordinate system φ : U V , where V is an open cell in Euclidean m-space such that φ(N U) is→ the subset of V satisfying ∩ xn+1 = xn+2 = ... = xm = 0, and φ restricted to U N is a local coordinate system for N around p. Then N is called a submanifold∩ of M. Example. The unit sphere x2 + y2 + z2 = 1 in R3. This is a differential manifold. So the unit sphere in R3 is a 2-dimensional submanifold of R3. 4 Many common examples of manifolds are submanifolds of Euclidean space. Remark. A submanifold is a subset of a manifold which is itself a man- ifold, but has smaller dimension. Manifold with boundary There is an important extension of the idea of a differential manifold as defined. This extension is motivated by the example of a closed disk D. An interior point p of D has a neighbourhood that is an open disk but a point q on the boundary has neighbourhood(in the disk) that is a half disk. Thus, if D is to be thought of as a differentiable manifold, two different kinds of coordinate neighbourhoods will have to be considered, according as the point in question is interior or not. Definition. A differential manifold of dimension n with boundary is a topological space M with a subspace N and a countable open covering U1,U2, ... with homeomorphisms φ1, φ2, ... satisfying the following conditions : 1) each set Ui of the given covering is either contained in M N, in which − case there is a homeomorphism φ : Ui Vi, where Vi is a solid open sphere → in n-space, or, otherwise there is a homeomorphism φ : Ui Vi where Vi is n 2 → a hemisphere of the form x < 1, xn 0, the set Ui N being mapped Pi=1 i ≥ ∩ on the subset of Vi for which xn = 0. 2) If Ui and Uj are two sets of the given covering and if φi and φj are the 1 homeomorphisms (just described) and if Ui Uj = φ, then φij = φi φ− is ∩ 6 ◦ j a differential map of φj(Ui Uj) onto φi(Ui Uj). ∩ ∩ Since homeomorphisms of the type φij must carry interior points into in- terior points and frontier points into frontier points, it is clear that if those Ui that meet N are restricted to N, they form a covering that defines a structure of an (n 1)-dimensional differentiable manifold on N. − If a manifold contains its own boundary, it is called, not surprisingly, a ”manifold with boundary”. The closed unit ball in Rn is a manifold with boundary, and its boundary is the unit sphere. The concept can be general- ized to manifolds with corners, by definition every point on a manifold has a 5 neighbourhood together with a homeomorphism of that neighbourhood with an open ball in Rn. Example. The solid sphere and the solid torus are differentiable 3- manifolds with boundaries, the boundaries, of course being the sphere and the torus, respectively. Tangent Vector and Tangent Vector Space Let M be a smooth n-dimensional manifold, x M is an arbitrary point, ∈ and x, the set of smooth functions at x. O ~ Definition. A tangent vector, Vx to M at a point x is a linear map ~ Vx : x R O −→ ~ f(x) Vxf −→ which satisfies the two axioms : ~ ~ ~ (1) Vx(αf + βg) = αVxf + βVxg; α, β R f, g x. ~ ~ ~ ∀ ∈ ∀ ∈ O (2) Vx(f.g) = f(x)Vxg + g(x)Vxf; f, g x. ∀ ∈ O ~ ~ If f x, Vx is a tangent vector, then Vxf is called the derivative of f ∈ O ~ along the vector Vx. Definition. The set of all tangent vectors at a point x denoted by TxM, together with addition and scalar multiplication defined by ~ ~ ~ ~ (αUx + βVx)(f) = αUx(f) + βVx(f) is a vector space called the tangent vector space. Theorem. Let dim M = n then the dimension of Tx0 M is equal to n at any point xo M. ∈ Proof. [2] or you can see any book of differential geometry. 6 Vector bundle Definition. A vector bundle of rank k (real or complex) over a manifold M n (called the base) is a manifold E (called the total space) and a projection π : E M such that → 1 (i) p M, Ep := π− (p) is a vector space of dimension k which is iso- ∀ ∈ k k morphic to R or C ; Ep is called the fibre over p.