1 Differentiable manifolds: definitions and examples
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1 1 Differentiable manifolds: definitions and examples Last updated: 14 (27) September 2016 Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set whose points can be parameterized by arrays of n independent variables. Introducing such a parametrization is called a coordinate system or a chart. It may happen that that it is not possible to have a single coordinate system for the whole manifold and we need to consider instead overlapping patches each with its own system of coordinates. Therefore we are forced to consider changes of coordinates. (For a particular patch, a coordinate system is not unique either.) Later we shall see that there is topology involved as well, because each manifold automatically carries a structure of a topological space. Before giving precise definitions, let us discuss first the fundamental idea of coordinates. What are coordinates? 1.1 Coordinates on familiar spaces. Example 1.1 (Standard coordinates on R n ). A point x R n is by definition an array (x 1,..., x n ). The numbers x i are called the standard coordinates on R n. (The superscripts are indices, not powers.) On R 2 and R 3, the standard coordinates are traditionally denoted x, y, z. Example 1.2 (General affine coordinates on R n ). Consider a change of variables: let y i be defined so that y 1 x 1 b 1... = A , y n x n b n where A is an invertible square matrix (det A 0) and b i some constants. Then the standard coordinates x i can be expressed back in terms of y j. Therefore there is a one-to-one correspondence x (y 1,..., y n ). The variables y j may be regarded as new affine coordinates on R n. Remark 1.1. The above is an operational definition of affine coordinates; the adjective affine refers to a particular form of the change of variables: namely, 1
2 linear inhomogeneous. If the constants b i above are all zero, b i = 0 for all i = 1,..., n, then y i are referred to as new linear coordinates. The geometric picture behind this is as follows. The space R n can be regarded as a vector space and its elements as vectors. Then the standard coordinates x i of x R n are the coefficients of the expansion of x, as a vector, over the standard basis vectors e 1 = (1, 0,..., 0),..., e n = (0, 0,..., 1), so x = x i e i. New linear coordinates y i on R n correspond to a choice of a new basis, say, g 1,..., g n ; they are the coefficients of the expansion over this new basis: x = y j g j. Therefore y i are related with the standard coordinates x i by a linear homogeneous transformation. The matrix A of this transformation is the transition matrix between the old basis e i and the new basis g j. To summarize, a linear coordinate system on R n is given by a choice of a basis. Also, R n can be regarded as an affine space meaning that its elements are regarded either as vectors (with the usual operations such as addition and multiplication by scalars) or as points, for which two new operations are introduced: subtraction of points, sending any two points P and Q in R n to a vector P Q, where P Q := Q P, and addition of a vector to a point, which sends a point P and a vector u to a point P + u, where P + u = Q such that P Q = u. An affine coordinate system on R n is given by a choice of a point O, called the origin, and a basis g i ; the coordinates of an arbitrary point x R n with respect to such a coordinate system (O, g i ) are the coefficients of the expansion of the radiusvector Ox over the chosen basis: Ox = y i g i (we use y i to avoid confusion with the standard coordinates of x). Linear coordinate systems, as opposed to general affine, are distinguished by fixing the origin as the point (0,..., 0); in this case a point x is identified with its own radius-vector. Example 1.3. Polar coordinates on R 2 ; spherical coordinates on R 3, and on R n for n > 3 (which can be defined by induction). For example, for R 2, x = (x, y) (r, θ) where x = r cos θ, y = r sin θ. Note: coordinates (r, θ) serve not for the whole R 2, but only for a part of it (an open subset). Recall that a set V R n is open is for each point x V there is an open ε-neighborhood entirely contained in V. (In a greater detail, there is ε > 0 such that V ε (x) V, where V ε (x) = {y R n x y < ε}. In other words, V ε (x) is an open ball of radius ε with center at x.) Remark 1.2. There are many reasons why open sets in R n are important. For us the main motivation is differential calculus, where one studies how the function changes if its argument is given a small increment, i.e., a given initial value of the argument is replaced by adding a small vector (which can point in an arbitrary direction). Therefore its is necessary to be able to consider a function on a whole neighborhood of any given point. So domains of definitions of functions have to be open if we wish to apply to them differential calculus. 2
3 Example 1.4 (Curvilinear coordinates on an open domain of R n ). In general, for an open domain U R n, differentiable functions y 1,..., y n defined on U make a system of curvilinear coordinates on U if it is possible to express back the standard coordinates x i in terms of y j, again by differentiable functions. So there a one-toone correspondence x (y 1,..., y n ) for x U differentiable in both directions. Then automatically the Jacobi matrix x 1 x... 1 y 1 y n x n x... n y 1 y n is invertible. (Polar and spherical coordinates are particular examples of curvilinear coordinates.) Remark 1.3. In this course we shall work with differentiable functions (of several variables). The standard notation for the class of functions possessing continuous derivatives of all orders k is C k. Though many statements remain valid for C k functions, for some finite differentiability order k, it is convenient to concentrate on functions of class C (those possessing derivatives of arbitrary order). They are called smooth. We shall use differentiable and smooth as synonyms, so for us all differentiable functions are supposed to have infinitely many derivatives. Affine coordinates on R n or curvilinear coordinates on open sets U R n have the property that one coordinate system works for the whole space, R n or U respectively. We shall consider now examples of coordinates on familiar objects where a single coordinate system is not enough and one needs to introduce coordinate patches. Example 1.5. The circle S 1 : x 2 +y 2 = 1 in R 2. A coordinate on the circle: u R, where u = x 1 y, and conversely x = 2u u 2 +1, y = u2 1 (check!). Good for all points of u 2 +1 the circle except N = (0, 1) (the north pole ). To have a coordinate system on S 1 covering the north pole, consider a different variable, denote it u R, such that u = x 1+y. It is defined at all points of the circle except for S = (0, 1) (the south pole ). In particular, u is defined for the north pole (where u = 0). Together the subsets where u or u are defined cover the whole S 1. On the intersection where both u and u are defined (it is S 1 \ {N, S}), the change of coordinates is u = 1 u (check!). Example 1.6. Similar stereographic coordinates can be defined for the unit 2- sphere S 2 in R 3 and, more generally, for the unit n-sphere S n R n+1. One needs to choose a point of the sphere S n as the center of projection and a plane (of dimension n) not containing this point as the plane of projection. Then the 3
4 projection map send an arbitrary point P S n (except for the center) to the point P on the projection plane defined as the intersection with the straight line passing through P and the projection center. This is clearly a one-to-one correspondence P P. Say, if one takes the north pole N = (0,..., 0, 1) as the center and the coordinate plane of the first n coordinates as the projection plane, the corresponding projection will be given by formulas similar to the above. Exercise: check that they have the form u N = x 1 z and x = 2u N u N 2 +1, z = u N 2 1 u N 2 +1, where P = (x, z) S n and P = (u N, 0), so that x and u are vectors in R n. Find the analogs for the projection from the south pole and calculate the change of coordinates. (Answer: u N = u S u S, if u 2 S R n corresponds to the projection from the south pole.) Example 1.7. Another way of introducing a coordinate on S 1 is to consider the polar angle θ. It is defined initially up to an integral multiple of 2π. To make it single-valued, we may restrict 0 < θ < 2π and thus we have to exclude the point (1, 0). To cover the whole circle, we may introduce θ so that π < θ < 3π and θ = θ for π < θ < 2π and θ = θ + 2π for 0 < θ < π. Example 1.8. Similarly, to obtain coordinates on S 2 R 3, one may use the angles θ, ϕ making part of the spherical coordinates on R 3. More generally, this can be done for S n R n+1 if one recalls the inductive construction of spherical coordinates on R n for any n. (Again, to be able to define such angular coordinates as single-valued functions, certain points have to be excluded from the sphere. To cover the whole S n, it will be necessary to consider several angular coordinate systems, each defined in a particular domain.) In each of the above examples, a familiar space (sphere) can be endowed with a collection of local coordinate systems. ( Local means that it covers only a part of the sphere.) The minimal number of coordinate patches here equals two; one can show that it is not possible to minimize it further, i.e., to cover the sphere by a single coordinate chart. Consider more examples. Example 1.9. Recall the notion of a projective space. The real projective space RP n is defined as the set of all straight lines through the origin in R n+1. In other words, it is the set of all one-dimensional subspaces of R n+1 considered as a vector space. So a point of RP n is a line in R n+1. Fix a hyperplane (a plane of dimension n) H R n+1 not through the origin. For example, it is possible to take the hyperplane x n+1 = 1. Each line through the origin O intersects H at a unique point, except for the lines parallel to H, which do not intersect H. The hyperplane H can be identified with R n by dropping the last coordinate x n+1 = 1. 4
5 Therefore the projective space RP n can be visualized as the ordinary ( affine ) n- dimensional space R n completed by adding extra points to it. 1 Notice that these extra points correspond to the straight lines through the origin in R n R n+1 considered as the coordinate hyperplane x n+1 = 0. Hence they themselves make RP n 1, and we have RP n = R n RP n 1 (so, by induction, RP n = R n R n 1... R 1 R 0 where R 0 is a single point). This construction introduces a coordinate system on the part RP n \RP n 1 of RP n. An inclusion RP n 1 RP n is equivalent to a choice of hyperplane H in R n+1. To cover by coordinates a different part of RP n, one has to choose a different H. It is not difficult to see that by taking as H the n + 1 coordinate hyperplanes x k = 1, where k = 1,..., n + 1, we obtain n + 1 coordinate systems covering together the whole RP n. Example The complex projective space CP n is defined similarly to RP n (with real numbers replaced by complex numbers). One can introduce coordinates into CP n in the same way as above. 1.2 Definition of a manifold. Fix a natural number n. We shall now give a series of definitions leading to the notion of an n-dimensional smooth manifold. Let X be an abstract set. Definition 1.1. A chart (or n-dimensional chart) for X is an bijective 2 map ϕ: V U where V R n is an open set in R n and U = ϕ(v ) X is a subset of X. Definition 1.2. An atlas (or n-dimensional atlas) for X is a collection of charts (ϕ α : V α U α ) such that the images cover the whole of X: X = α U α 1 Extra points that one adds to the affine space R n in order to obtain the projective space RP n are often referred to as points at infinity. Indeed, a line in R n+1 parallel to H may be visualized as the limiting position of lines intersecting H when the point of intersection goes farther and farther away from some point in H taken as an origin. 2 Recall that a map ϕ is bijective if it is both injective and surjective; injective means that ϕ(a) = ϕ(b) implies a = b; and surjective means onto, i.e., that every element of the codomain is the image of some element of the domain. Being bijective is equivalent to being invertible. 5
6 where U α = ϕ α (V α ). A chart ϕ: V U gives a one-to-one correspondence between points x U X and arrays (x 1,..., x n ) V R n given by the maps ϕ and : X U x = ϕ(x 1,..., x n ) (x 1,..., x n ) V R n. We call the n real numbers x 1,..., x n, the coordinates of a point x U X. Formally, (x 1,..., x n ) = (x). For this reason we also call a chart for X, a coordinate system on X. It is also called a local coordinate system on X to emphasize that makes sense only for a subset U X. It is convenient to think that the R n s for different charts are different copies 3 of the space R n and to denote them R n (α), so that V α R n (α). Consider sets U α and U such that U α U. To the intersection U α U correspond subsets α (U α U ) V α and (U α U ) V. Any point x U α U has two coordinate descriptions: α (x) = (x 1 α,..., x n α) and (x) = (x1,..., xn ). Therefore there is an invertible map α ϕ : (U α U ) α (U α U ), (x 1,..., x n ) (x 1 α,..., x n α), which we call the change of coordinates between charts ϕ a and ϕ. Definition 1.3. Two charts (ϕ α : V α U α ) and (ϕ : V U ) are compatible if both sets α (U α U ) R n (α) and ϕ 1 (U α U ) R n () are open and the mutually inverse changes of coordinates and α ϕ : (U α U ) α (U α U ) ϕ α : α (U α U ) (U α U ) are given by differentiable (smooth) functions. Definition 1.4. An atlas A = (ϕ α : V α U α ) is differentiable or smooth if it consists of (pairwise) compatible charts. (That means that all the sets α (U α U ) are open and all the changes of coordinates α ϕ are given by differentiable functions.) 3 Think about a geographical atlas. Geographical maps for the Earth corresponds to mathematical charts for a set X. Different pages correspond to different R n (α) s. 6
7 Recall that by our convention smooth or differentiable means C. Note that the first part of the condition, that the sets α (U α U ) R n (α) are open for all α, is necessary for the second part, that the changes of coordinates are smooth, to make sense. Coordinate changes are invertible smooth maps between open subsets of (different copies of) R n. Definition 1.5. We say that the set X is an n-dimensional differentiable (or smooth) manifold if X is endowed with a smooth n-dimensional atlas. The number n is called the dimension of X, n = dim X. It is important to understand that a manifold is not just a set, but rather a pair (X, A) consisting of a set X and a smooth atlas A on X. With a common abuse of language, we speak of a manifold X, where a certain atlas is implicitly understood. (An extra refinement appears below, after the introduction of the notion of equivalent atlases.) Besides X, Y and Z, other traditional letters for denoting manifolds are M, N, P, and Q. The dimension of a manifold M is often indicated by a superscript, e.g., we write M = M n or X k indicating that dim M = n and dim X = k. Suppose on one set M, two smooth atlases are defined, so we have two smooth manifolds, (M, A 1 ) and (M, A 2 ). For example, for the circle S 1 (or the sphere S n ) we can consider the atlas consisting of the two stereographic charts as above or an atlas constructed using angular coordinates. Are these manifold structures the same? Definition 1.6. Two smooth atlases A 1 and A 2 on the same set M are equivalent if their union A 1 A 2 is also a smooth atlas. This is equivalent to saying that any chart from A 1 and any chart from A 2 are compatible. Manifolds (M, A 1 ) and (M, A 2 ) (with the same underlying set M) are regarded the same manifold if the atlases A 1 and A 2 are equivalent. Of course, the atlases for S 1 defined by using stereographic coordinates and by using the polar angles are equivalent. For illustrative purposes, let us give an example of non-equivalent atlases. Example Consider two atlases on R each consisting of a single chart: for A 1 = (φ: R R), φ(x) = x (the identity map), and for A 2 = (ψ : R R), ψ(x) = x 3. Check that these atlases are NOT equivalent. 7
8 We see that, strictly speaking, a manifold should be regarded as a pair consisting of a set and the equivalence class of a smooth atlas (rather than a single atlas). A way around that is to introduce maximal atlases as the unions of all atlases that are pairwise equivalent. Then a manifold can be unambiguously seen as a pair consisting of a set and a maximal atlas for this set. This is convenient for theoretical considerations. For practical purposes, we always work with some particular fixed atlas (in a given equivalence class) and try to minimize the number of charts in it. Remark 1.4. We shall normally omit the adjective and call differentiable manifolds simply manifolds because no other kinds of manifolds will be considered 4. It is worth at least to mention some options. If no smoothness condition is imposed, but the changes of coordinates are assumed to be just continuous maps (homeomorphisms) of open subsets of R n, such a version of manifolds is called topological manifolds. They properly belong to topology, not differential geometry. On the opposite extreme, one may require that the changes of coordinates are given by analytic rather than C functions. (A function is analytic if it can be written as the sum of a power series.) Such a version of manifolds is called real-analytic manifolds. Many of our examples are in fact real-analytic. Also, one may consider open domains of C n instead of R n, introducing complex numbers as coordinates, and require that the changes of coordinates are given by holomorphic (i.e., complexanalytic) functions. In such a way complex manifolds are defined. Any complex manifold of complex dimension n is automatically a smooth (in fact, real-analytic) manifold of dimension 2n. Some of our examples will be, in fact, complex manifolds. 1.3 Examples of manifolds Example Consider S 1. Introduce two charts: ϕ N : R S 1 and ϕ S : R S 1, where ( 2uN ϕ N : u N u2 N 1 ) u 2 N + 1, u 2 N + 1 and ϕ S : u S ( 2uS u 2 + 1, 1 ) u2 S u 2 S One can define differentiable manifolds of a particular class of smoothness C k, for a fixed given k (i.e., k continuous derivatives), but we shall not do that. 8
9 (see Example 1.5). They correspond to the stereographic projections from the north pole N = (0, 1) and the south pole S = (0, 1) respectively. We have V N = R, U N = S 1 \{N}, V S = R, U S = S 1 \{S}, and U N U S = S 1 \{N, S}. For the change of coordinates we obtain N ϕ S : R \ {0} R \ {0}, u S u N = 1 u S (check!). Therefore it is smooth, and we conclude that S 1 with this atlas is a smooth manifold of dimension 1. Example In the same way we can obtain a smooth atlas consisting of two charts on any sphere S n R n+1. (Find the explicit formulas and check the smoothness!) This makes S n a smooth manifold of dimension n. Example A point of RP n can be identified with a non-zero vector v = R n+1 considered up to a non-zero scalar factor, v kv, k 0. The coordinates of v considered up to a factor are written as (x 1 :... : x n : x n+1 ) and traditionally called the homogeneous coordinates on RP n. (They are not coordinates in the true sense, because are defined only up to a factor.) The construction described in Example 1.9 gives a chart ϕ: R n RP n, (y 1,..., y n ) (y 1 :... : y n : 1) with the image V = V (n+1) RP n specified by the inequality x n+1 0 in homogeneous coordinates. The inverse map : V R n is ( x (x 1 :... : x n : x n+1 1 ) x,..., x n ). n+1 x n+1 Similarly we can define other charts ϕ (k), k = 1,..., n, corresponding to the choice of x k as a non-zero homogeneous coordinate (i.e., to the choice {x k = 1} as the hyperplane H in Example 1.9). Together with ϕ (n+1) they make an atlas for RP n consisting of n + 1 charts. It is smooth. (Write down the explicit formulas and check!) Hence, RP n with this atlas becomes an n-dimensional smooth manifold. Coordinates in any of these charts are traditionally called the inhomogeneous coordinates on RP n. Example Acting similarly for CP n, we obtain the n + 1 charts ϕ (k) : C n CP n again giving a smooth atlas. Hence CP n has the structure of a 2n-dimensional manifold. (Each complex coordinate gives two real coordinates.) 9
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