Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M.

Save this PDF as:
Size: px
Start display at page:

Download "Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M."

Transcription

1 5 Vector fields Last updated: March 12, Definition and general properties We first need to define what a vector field is. Definition 5.1. A vector field v on a manifold M is map M T M such that for all x M, v(x) T x M. Vector fields are traditional denoted by boldface letters such as v, u or w, or by capital letters such as X, Y or Z. The set of all vector fields on M is denoted X(M). Proposition 5.1. The set X(M) of all vector fields on a manifold M is a vector space over R. Moreover, vector fields can be multiplied by functions, so that (fu)(x) = f(x) u(x), f(v + u) = fu + fv, f(λu) = λfu, (fg)v = f(gv), (f + g)v = fv + gv, and 1v = v. for arbitrary vector fields u, v, functions g, g, and a constant λ. Proof. Obvious. 5.2 Digression: tangent vectors as derivations There is an alternative viewpoint at tangent vectors. Let M = M n be a manifold. Consider a vector v T x0 M. Suppose γ : ( ε, ε) M is a curve such that x 0 = γ(0) = x(0) and v = dx dt at t = 0. Definition 5.2. For an arbitrary function f : M R the number v f := d f(x(t)). dt t=0 (the derivative at t = 0) is called the derivative of f along v T x0 M. 1

2 Comparing it with the definition of the differential, we see that v f = df(x 0 )(v). If x 1,..., x n are coordinates near x and v i, i = 1,..., n are the components of v, we have n f v f = x (x 0) v i. i i=1 Proposition 5.2. The operation v : C (M) R satisfies the following properties: linearity over R and the Leibniz rule Proof. Immediate. v (fg) = v f g(x) + f(x) v g. Note that the map sending a function f C (M) to the number f(x 0 ) R is a homomorphism, called the evaluation homomorphism at x 0. We denote it ev x0. There are fundamental algebraic notions: Definition 5.3. For a given homomorphism of algebras α: A 1 linear map D : A 1 A 2 is called a derivation over α if A 2, a D(ab) = D(a) α(b) + α(a) D(b) for all a, b A 1. In the special case of a single algebra A 1 = A 2 = A and α = id (the identity map), a derivation over it is simply called a derivation of the algebra A. Hence for any v T x0 M the operation v : C (M) R is a derivation over the evaluation homomorphism at x 0 M. Remark 5.1. It is possible to consider v on C (V ) for any open set V M s.t. x 0 V, and it is a derivation C (V ) R over ev x0 as well. It turns out that all the derivations of the algebra of functions on a manifold to numbers are of the form v. Let us first explore the case of R n. Theorem 5.1. Let x 0 R n. For an arbitrary derivation D : C (R n ) R over the evaluation homomorphism ev x0 there is a vector v T x0 R n such that D = v. 2

3 The proof uses the following simple but fundamental statement: Lemma 5.1 (Hadamard s Lemma). For any smooth function f C (R n ) and any x 0 R n there is an expansion f(x) = f(x 0 ) + n (x i x i 0)g i (x) i=1 where g i C (R n ) are smooth functions. Proof. Consider the segment joining x and x 0 and write f(x) = f(x 0 ) d dt f(x 0 + t(x x 0 )) dt = f(x 0 ) + (x i x i 0) 1 0 f x i (x 0 + t(x x 0 )) dt Corollary 5.1. There is an expansion f(x) = f(x 0 ) + n i=1 (x i x i 0) f x i (x 0) n (x i x i 0)(x j x j 0)g ij (x) (1) i,j=1 where g ij C (R n ). Proof. By iterating the previous expansion we arrive at f(x) = f(x 0 ) + n (x i x i 0)a i i=1 n (x i x i 0)(x j x j 0) g ij (x) i,j=1 where a i R are numbers and g ij C (R n ), functions. Apply partial derivative at x x i 0 and obtain a i = f (x x i 0 ). Remark 5.2. Corollary 5.1 is a form of the Taylor expansion to the first order. The Taylor expansion to any order N can be similarly deduced from Hadamard s Lemma, with the remainder of order N + 1 being the so-called remainder in the integral form. Now we can prove the main theorem. 3

4 Proof of Theorem 5.1. Consider a point x 0 R n. (We are keeping x as a running point.) Let D : C (R n ) R be a derivation over the evaluation at x 0. Apply D to the expansion (1). First note that derivations kill constants; indeed, D(1) = D(1 1) = D(1) D(1) = 2D(1), hence D(1) = 0 and then D(c) = D(c 1) = 0 for any c R. Therefore we obtain D(f) = D(x i ) f x (x 0)+ i i 1 ( ) D(x i )(x j 2 0 x j 0) g ij (x 0 )+(x i 0 x i 0)D(x j ) g ij (x 0 )+(x i x i 0)(x j x j 0) D(g ij ) = ij for v = (v 1,..., v n ) where v i = D(x i ). i D(x i ) f x i (x 0) = v f Theorem 5.1 can be immediately transferred to open domains of a manifold U M admitting a single coordinate system: any derivation D : C (U) R ev x0, where x 0 U M, is of the form D = v for some v T x0 M if there is a chart ϕ: V U, where V R n. This can be made slightly more formal by introducing the so-called germs of functions at a given point. Consider functions defined on open neighborhoods U M of x 0. A function f defined on U and a function g defined on U are said to be equivalent if there is an open neighborhood U of x 0 such that U U U and f U = g U. The equivalence class of a (local) function f defined near x 0 is called its germ. Germs at x 0 make an algebra, notation: F x0 ; and there an evaluation homomorphism ev x0 : F x0 R. We arrive at the following theorem. are in one-to-one corre- Theorem 5.2. The derivations F x0 R over ev x0 spondence with the tangent vectors v T x0 M. It is a common practice to identify vectors v with the corresponding derivations v. For example, the coordinate basis vectors e i = x are identified with the partial derivatives i =. x i x i Now what about global functions, i.e., the algebra C (M)? 4

5 Theorem 5.3. Every derivation C (M) R over ev x0 a tangent vector v T x0 M. has the form v for Proof. Consider a derivation D : C (M) R over the evaluation at x 0. We want to show that it is possible to apply D to germs of functions at x 0. Suppose we have a function defined only locally near x 0. We know that any such function can be extended to the whole manifold without changing it on a smaller neighborhood of x 0 (by using bump functions). Therefore any germ at x 0 is the germ of a global function. To define the action of D on a germ by applying it to a global function representing it, we need to show that if two functions f, g C (M) coincide on a neighborhood of x, then Df = Dg. This is equivalent to the following property: if a function f C (M) vanishes on an open neighborhood of x 0, then it is annihilated by D. We claim that if a function f C (M) vanishes on an open neighborhood U of x 0, then it can written as the product of functions in C (M) vanishing at x 0. Indeed, consider a function h C (M) such that h = 0 on W U and h = 1 on M \ U (it is 1 h where h is a suitable bump function). Then hf = f. Note that both f and h vanish at x 0, and f is the product of such functions. Now we see that every derivation D : C (M) R over ev x0 annihilates such an f (because Df = D(hf) = Dh f(x 0 ) + h(x 0 ) Df = = 0). Hence all such derivations can be unambiguously applied to germs at x 0. We arrive at an isomorphism between the two spaces: the space of all derivations C (M) R over ev x0 and the space of all derivations F x0 R over ev x0. Now we can use Theorem Commutator of vector fields Everything what is said below applies to vector fields defined not necessarily on the whole manifold M, but on an open subset U M. We shall speak of vector fields on M for the simplicity of notation. We have established that a tangent vector v at a point x M defines a derivation at x, i.e., a linear map v : C (M) R satisfying v (fg) = v f g(x) + f(x) v g for all f, g C (M), and is frequently identified with this map. Moreover, v is the general form of a derivation D : C (M) R over ev x. Now, if we consider a vector field instead of a single vector, that means that we allow x in the formula above to vary. For a vector field v X(M), 5

6 we arrive at the linear map v : C (M) C (M) of the algebra C (M) to itself satisfying v (fg) = v f g + f v g. Recall that in the previous subsection, such linear operators on algebras were called derivations of algebras. (See Definition 5.3. The relation with the general notion of a derivation over an algebra homomorphism is as follows: a derivation of an algebra is the same as a derivation over the identity homomorphism of this algebra. The set of all derivations of an algebra A will be denoted Der A. Corollary 5.2 (From Theorem 5.3). The space of all vector fields on M can be identified with the space of all derivations of the algebra C (M) : X(M) = Der C (M). When we write vector fields in local coordinates, we have v = v i (x) x i for v = v i (x) x x i, and very frequently we identify v with v writing simply v = v i (x) x i. There is a convenient abbreviation i = / x i, so one can write v = v i i. Recall the following notion: for two linear operators A and B on a vector space V, their commutator is denoted by [A, B] and defined as [A, B] = AB BA. (Here the product of operators is the composition: AB = A B.) Theorem 5.4. The commutator of vector fields considered as linear operators on the algebra of functions is again a vector field. Proof. Calculation in coordinates. Suppose we are given X, Y X(M) and in some local coordinates X = X i i, Y = Y i i. For an arbitrary function f we have X(Y f) = X i i (Y j j f) = X i i Y j j f + X i Y j 2 ijf 6

7 (where we identify vector fields with operators on functions). Here 2 ijf = 2 f x i x j. In the same way we have Y (Xf) = Y i i X j j f + Y i X j 2 ijf. We can rename indices in the second term to get Y (Xf) = Y i i X j j f + Y j X i 2 jif = Y i i X j j f + Y j X i 2 ijf, where we have used the commutativity of the second partial derivatives. Therefore we obtain (XY Y X)f = X(Y f) Y (Xf) = X i i Y j j f Y i i X j j f = (X i i Y j Y i i X j ) j f. We conclude that the commutator of vector fields X and Y is indeed a vector field given by ) [X, Y ] = (X i i Y j Y i i X j j (2) in local coordinates. We have obtained a binary operation on vector fields, called their commutator or (sometimes) the Lie bracket. An explicit formula is coordinates is equation (2). The following form of this formula is convenient for practical calculations: [X, Y ] = ( X Y j Y X j ) j. (3) Remark 5.3. The above expression has the appearance of taking the derivative of the vector field Y along the vector field X minus the same with X and Y swapped. Here taking the derivative of a vector field along another vector field is understood naively as taking the derivative of the components. Unfortunately, this makes sense only in a fixed coordinate system because X Y i does not transform to a different coordinate system as a component of a vector. However, the difference X Y i Y X i does, so it makes good sense independent of coordinates. A rectification of the naive notion of a derivative of a vector field leads to the concept of a so-called covariant derivative X Y. Such an object requires introducing an extra piece of data on a manifold called a connection. This is the crucial difference with commutator of vector fields, which is a natural operation on geometric objects on manifolds in the sense that it does not require anything but a manifold structure and can be expressed by a formula valid in arbitrary coordinates. 7

8 Remark 5.4. As one can see from the calculation, the product (composition) XY of two vector fields X, Y as operators on functions is no longer a vector field, but is a differential operator of the second order. Hence the significance of the fact that the commutator [X, Y ] is a vector field. This can be put in a larger perspective by noting that for two differential operators of orders p and q (acting on functions), their composition is a differential operator of order p + q, but the commutator is an operator of order p + q 1. Our proof of Theorem 5.4 was based on a direct coordinate calculation. A more elucidating approach is possible. For an arbitrary algebra A consider the space of all derivations Der A, i.e., the linear operators D : A A satisfying the Leibniz identity for all elements a, b A. D(ab) = D(a) b + a D(b) Theorem 5.5. The commutator of derivations is a derivation. Proof. Consider two derivations D 1 and D 2. We have D 1 (D 2 (ab)) = D 1 ( D2 (a) b + a D 2 (b) ) = D 1 (D 2 (a)) b + D 2 (a) D 1 (b)+ and similarly D 1 (a) D 2 (b) + a D 1 (D 2 (b)) D 2 (D 1 (ab)) = D 2 (D 1 (a)) b + D 1 (a) D 2 (b) + D 2 (a) D 1 (b) + a D 2 (D 1 (b)). After subtracting, the cross terms such as D 1 (a) D 2 (b) are cancelled and we arrive at [D 1, D 2 ](ab) = D 1 (D 2 (ab)) D 2 (D 1 (ab)) = D 1 (D 2 (a)) b D 2 (D 1 (a)) b + a D 1 (D 2 (b)) a D 2 (D 1 (b)) = Hence [D 1, D 2 ] is a derivation as claimed. [D 1, D 2 ](a) b + a [D 1, D 2 ](b). Now we see that Theorem 5.4 follows from Theorem 5.5 if we identify the space X(M) with Der A for the algebra A = C (M). Theorem 5.6. The commutator of vector fields has the following properties: 8

9 1. bilinearity over numbers: [cx, Y ] = c[x, Y ], [X + Y, Z] = [X, Z] + [Y, Z], (and the same w.r.t. the second argument); 2. antisymmetry: [X, Y ] = [Y, X] ; 3. Jacobi identity: for all X, Y, Z X(M) and c R. [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = 0, Proof. Bilinearity and antisymmetry are obvious from the definition. Let us check the Jacobi identity. By expanding the commutators we have [X, [Y, Z]] + [Z, [X, Y ]] + [Y, [Z, X]] = as claimed. X(Y Z ZY ) (Y Z ZY )X + Z(XY Y X) (XY Y X)Z + Y (ZX XZ) (ZX XZ)Y = XY Z XZY Y ZX + ZY X + ZXY ZY X XY Z + Y XZ + Y ZX Y XZ ZXY + XZY = 0 Definition 5.4. A vector space endowed with a bilinear antisymmetric operator satisfying the Jacobi identity is called a Lie algebra. Therefore the space X(M) with the operation of commutator is a Lie algebra. Remark 5.5. The proof of Theorem 5.6 uses nothing but the associativity of the composition of linear operators. An analog of Theorem 5.6 holds therefore for commutator [a, b] = ab ba in any associative algebra A, in particular, for the algebra of all linear operators on a vector space, or for any subspace of it that is closed under commutator. (Vector fields give a good example: as we know, the space X(M) is not closed under composition, but is closed under commutator.) Lie algebras play fundamental role in many areas of mathematics. Typical examples of Lie algebras, besides the Lie algebras of vector fields X(M), include various matrix Lie algebras, where the operation is the matrix commutator. See examples below. 9

10 Remark 5.6. Vector fields can also be multiplied by functions, but the commutator of vector fields is not bilinear w.r.t. this multiplication (one cannot take functions out). Check that [X, fy ] = f[x, Y ] + (Xf) Y, where Xf = X f = X i i f is the extra term. Example 5.1. Check that the following subspaces of the space of all n n matrices are closed under the commutator: the space of all antisymmetric matrices o(n); the space of all trace-free matrices sl(n); (for matrices with complex entries) the space of all anti-hermitian matrices, i.e., satisfying A = A, where A = ĀT, denoted u(n); the space of all upper-triangular matrices (unlike the previous examples, it is already closed under the matrix product). Therefore all these spaces give examples of Lie algebras. Remark 5.7. The Lie matrix algebras appearing in Example 5.1 such as o(n), sl(n), u(n) (and similar) are related with the matrix Lie groups such as O(n), SL(n) and U(n). By the tradition, the lower case Gothic letters (i.e., the German typeface Fraktur ) are used for denoting Lie algebras. Example 5.2. The ordinary Euclidean three-space becomes a Lie algebra w.r.t. the operation of vector product (or cross-product): (u, v) u v, defined, e.g., via a symbolic determinant. It is not easy to establish the Jacobi identity directly. However, one can check that the following linear transformation 0 u 3 u 2 u = (u 1, u 2, u 3 ) u 3 0 u 1 u 2 u 1 0 mapping vectors from R 3 to the antisymmetric matrices in o(3) is an isomorphism of the two vector spaces and it maps the vector product of vectors in R 3 to the commutator of matrices. (It is sufficient to check that for the standard basis vectors i, j, k. This map is nothing but u X u where 10

11 X u (v) = u v. Every antisymmetric operator X on R 3 is the vector product with a unique vector u R 3, X = X u, which is known as the Darboux vector for X o(3).) In this way the Jacobi identity for vector product on R 3 follows from that for matrix commutator and we see that the space R 3 with vector product is another realization of the Lie algebra o(3). 5.4 The flow of a vector field Vector fields on a manifold have the following geometrical interpretation. We have a manifold (or its open subset) and to each point of it a tangent vector is attached. If we imagine our manifold as sitting in some large R N, we have a picture of tangent arrows at each point 1. Interpreting a tangent vector as the velocity of a curve, we have curves filling our manifold (or its open domain) so that through every point passes a unique curve and its velocity at this point is exactly the given vector at this point. These curves are called the integral curves of a vector field. (We elaborate this below.) We can imagine a flow of some fluid flowing on our manifold so that the velocity of the flow at each point is given by the vector attached to that point. (In particular, this is a stationary flow, in the sense that the velocities of particles traveling through any given point are the same and do not depend on the time, so the velocity is a function of a point only.) Such a hydrodynamic interpretation of vector fields as the velocity fields is very important. Now we shall elaborate it and in particular define the flow of a vector field as a precise mathematical notion. If u = u(x) is a vector field on M, we can associate with it the following ordinary differential equation on M, which becomes a system of ODEs when written in coordinates: dx = u(x). (4) dt Here x = x(t) and the parameter t (the time) runs over some interval, e.g., t ( ε, ε). In coordinates, dx i dt = ui (x), (5) where i = 1,..., n. This is a system of non-linear (in general) ordinary differential equations, with the RHS not depending on the time explicitly. (Such systems are called autonomous.) We shall make use of the two main facts concerning such systems. Firstly, if an initial value x(t 0 ) = x 0 is fixed for some moment of time t 0, there is a unique solution x = x(t) with this initial value for t in some interval 1 Embedding the manifold into R N is used only for better visualization. What follows does not require any such embedding. 11

12 around t = t 0 ( the existence and uniqueness theorem for solutions of ODEs ). The solutions x = x(t) are called the trajectories or the integral curves of the vector field u. Secondly, this unique solution of our ODE with a given initial value x(t 0 ) = x 0 depends smoothly on x 0 M ( the smooth dependence on initial value theorem ). It follows that we have a well-defined smooth map g t : M M, g t : x 0 g t (x 0 ) = x(t), for each t, where x(t) is the solution of (4) with the initial value x(0) = x 0. Here t belongs to some interval around zero, depending, in principle, on x 0. To avoid complicated notation we shall neglect this fact and write all the formulas as if t can take any value. The uniqueness of solution implies that the family of maps g t : M M has the following properties: g 0 = id (6) (indeed, this simply restates that g 0 (x 0 ) = x(0) = x 0 for any x 0 M taken as the initial value for x(t) at t = 0); g t = g 1 t (7) (indeed, this simply says that if we travel along a trajectory from any given point backward in time for the time interval t and then take the result as an initial value and travel in the forward direction for the same t, we shall return to the original point; and the same if we do it other way round: first moving forward and then, back); g t+s = g t g s (8) (this means that, starting from any point, if we travel along a trajectory for the time interval s and then, t, it is the same as to travel for the interval t + s; again follows from the uniqueness of solution). Any family of transformations of a manifold M satisfying (6), (7), (8), which are automatically invertible due to (7), is called a one-parameter group of transformations (or diffeomorphisms) of M or, shortly, a flow on M. We see that any vector field on M gives rise to a flow, which is called the flow of a vector field (or: generated by a vector field), with the vector field called as the generator of a flow. Finding the flow for a given vector field u is the same as solving equation (4) for all initial values. Example 5.3. Let M = R n and u(x) = a (a constant vector). Then (4) will be dx dt = a, 12

13 which has the general solution x = at + x 0, where x 0 = x(0). Hence the flow g t consists of parallel shifts of all points in the direction of the constant vector a: g t : x x + at for any t R. The trajectories are the straight lines parallel to a. Example 5.4. Let M = R 2 and suppose a vector field X is given in Cartesian coordinates as X = ye x + xe y. To find its flow we have to solve the system { ẋ = y ẏ = x (the time derivative denoted by the dot). This can be written in the matrix form as ( ) ( ) ( ) d x 0 1 x =. dt y 1 0 y The solution is given by the matrix exponential: where A = x = e At x 0 ( ) Since in our case A 2 = E, A 3 = A, A 4 = E, etc. (E is the identity matrix), we have ( ) e At cos t sin t =. sin t cos t Therefore the flow g t : R 2 R 2 is ( ) ( ) ( ) x cos t sin t x g t : y sin t cos t y i.e., is the rotations around the origin through angle t. The trajectories are the circles with the center at the origin, and the origin itself (the whole trajectory is one point). Considering flows of vector fields allows to relate Lie algebras with Lie groups. For example, the examples of matrix Lie algebras introduced above can be associated with the corresponding examples of groups of transformations (e.g., antisymmetric matrices with orthogonal transformations, etc.). Commutator of vector fields can be given an interpretation in terms of the corresponding flows. One can 13,

14 see that, for vector fields X and Y, the commutator [X, Y ] at a point x can be obtained by considering the group commutator h s g t h s g t applied to x and expanded over t, s to the second order neglecting t 2 and s 2. The terms proportional t and s naturally cancel and the only remaining term will be proportional to ts, and this is exactly ts[x, Y ](x) (check!). 14

THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.)

THEODORE VORONOV DIFFERENTIABLE MANIFOLDS. Fall Last updated: November 26, (Under construction.) 4 Vector fields Last updated: November 26, 2009. (Under construction.) 4.1 Tangent vectors as derivations After we have introduced topological notions, we can come back to analysis on manifolds. Let M

More information

7 Curvature of a connection

7 Curvature of a connection [under construction] 7 Curvature of a connection 7.1 Theorema Egregium Consider the derivation equations for a hypersurface in R n+1. We are mostly interested in the case n = 2, but shall start from the

More information

CALCULUS ON MANIFOLDS

CALCULUS ON MANIFOLDS CALCULUS ON MANIFOLDS 1. Manifolds Morally, manifolds are topological spaces which locally look like open balls of the Euclidean space R n. One can construct them by piecing together such balls ( cells

More information

1 Differentiable manifolds and smooth maps

1 Differentiable manifolds and smooth maps 1 Differentiable manifolds and smooth maps Last updated: April 14, 2011. 1.1 Examples and definitions Roughly, manifolds are sets where one can introduce coordinates. An n-dimensional manifold is a set

More information

0.1 Diffeomorphisms. 0.2 The differential

0.1 Diffeomorphisms. 0.2 The differential Lectures 6 and 7, October 10 and 12 Easy fact: An open subset of a differentiable manifold is a differentiable manifold of the same dimension the ambient space differentiable structure induces a differentiable

More information

1 Introduction: connections and fiber bundles

1 Introduction: connections and fiber bundles [under construction] 1 Introduction: connections and fiber bundles Two main concepts of differential geometry are those of a covariant derivative and of a fiber bundle (in particular, a vector bundle).

More information

CALCULUS ON MANIFOLDS

CALCULUS ON MANIFOLDS CALCULUS ON MANIFOLDS 1. Introduction. Crash course on the Multivariate Calculus 1.1. Linear algebra. Field of real numbers R and its completeness. Real line R 1 and real spaces R n. Linear functionals,

More information

1 Differentiable manifolds and smooth maps. (Solutions)

1 Differentiable manifolds and smooth maps. (Solutions) 1 Differentiable manifolds and smooth maps Solutions Last updated: March 17 2011 Problem 1 The state of the planar pendulum is entirely defined by the position of its moving end in the plane R 2 Since

More information

CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =

CALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.

More information

5 Constructions of connections

5 Constructions of connections [under construction] 5 Constructions of connections 5.1 Connections on manifolds and the Levi-Civita theorem We start with a bit of terminology. A connection on the tangent bundle T M M of a manifold M

More information

THEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009

THEODORE VORONOV DIFFERENTIAL GEOMETRY. Spring 2009 [under construction] 8 Parallel transport 8.1 Equation of parallel transport Consider a vector bundle E B. We would like to compare vectors belonging to fibers over different points. Recall that this was

More information

Topics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map

Topics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map Topics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map Most of the groups we will be considering this semester will be matrix groups, i.e. subgroups of G = Aut(V ), the group

More information

Sec. 1.1: Basics of Vectors

Sec. 1.1: Basics of Vectors Sec. 1.1: Basics of Vectors Notation for Euclidean space R n : all points (x 1, x 2,..., x n ) in n-dimensional space. Examples: 1. R 1 : all points on the real number line. 2. R 2 : all points (x 1, x

More information

Math 147, Homework 1 Solutions Due: April 10, 2012

Math 147, Homework 1 Solutions Due: April 10, 2012 1. For what values of a is the set: Math 147, Homework 1 Solutions Due: April 10, 2012 M a = { (x, y, z) : x 2 + y 2 z 2 = a } a smooth manifold? Give explicit parametrizations for open sets covering M

More information

1 Differentiable manifolds and smooth maps. (Solutions)

1 Differentiable manifolds and smooth maps. (Solutions) 1 Differentiable manifolds and smooth maps Solutions Last updated: February 16 2012 Problem 1 a The projection maps a point P x y S 1 to the point P u 0 R 2 the intersection of the line NP with the x-axis

More information

2 Constructions of manifolds. (Solutions)

2 Constructions of manifolds. (Solutions) 2 Constructions of manifolds. (Solutions) Last updated: February 16, 2012. Problem 1. The state of a double pendulum is entirely defined by the positions of the moving ends of the two simple pendula of

More information

10. Smooth Varieties. 82 Andreas Gathmann

10. Smooth Varieties. 82 Andreas Gathmann 82 Andreas Gathmann 10. Smooth Varieties Let a be a point on a variety X. In the last chapter we have introduced the tangent cone C a X as a way to study X locally around a (see Construction 9.20). It

More information

Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course

Matrix Lie groups. and their Lie algebras. Mahmood Alaghmandan. A project in fulfillment of the requirement for the Lie algebra course Matrix Lie groups and their Lie algebras Mahmood Alaghmandan A project in fulfillment of the requirement for the Lie algebra course Department of Mathematics and Statistics University of Saskatchewan March

More information

b) The system of ODE s d x = v(x) in U. (2) dt

b) The system of ODE s d x = v(x) in U. (2) dt How to solve linear and quasilinear first order partial differential equations This text contains a sketch about how to solve linear and quasilinear first order PDEs and should prepare you for the general

More information

7. Baker-Campbell-Hausdorff formula

7. Baker-Campbell-Hausdorff formula 7. Baker-Campbell-Hausdorff formula 7.1. Formulation. Let G GL(n,R) be a matrix Lie group and let g = Lie(G). The exponential map is an analytic diffeomorphim of a neighborhood of 0 in g with a neighborhood

More information

4.7 The Levi-Civita connection and parallel transport

4.7 The Levi-Civita connection and parallel transport Classnotes: Geometry & Control of Dynamical Systems, M. Kawski. April 21, 2009 138 4.7 The Levi-Civita connection and parallel transport In the earlier investigation, characterizing the shortest curves

More information

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that

3.1. Derivations. Let A be a commutative k-algebra. Let M be a left A-module. A derivation of A in M is a linear map D : A M such that ALGEBRAIC GROUPS 33 3. Lie algebras Now we introduce the Lie algebra of an algebraic group. First, we need to do some more algebraic geometry to understand the tangent space to an algebraic variety at

More information

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))

fy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f)) 1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical

More information

Reminder on basic differential geometry

Reminder on basic differential geometry Reminder on basic differential geometry for the mastermath course of 2013 Charts Manifolds will be denoted by M, N etc. One should think of a manifold as made out of points (while the elements of a vector

More information

Math 147, Homework 5 Solutions Due: May 15, 2012

Math 147, Homework 5 Solutions Due: May 15, 2012 Math 147, Homework 5 Solutions Due: May 15, 2012 1 Let f : R 3 R 6 and φ : R 3 R 3 be the smooth maps defined by: f(x, y, z) = (x 2, y 2, z 2, xy, xz, yz) and φ(x, y, z) = ( x, y, z) (a) Show that f is

More information

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY

ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up

More information

MANIFOLD STRUCTURES IN ALGEBRA

MANIFOLD STRUCTURES IN ALGEBRA MANIFOLD STRUCTURES IN ALGEBRA MATTHEW GARCIA 1. Definitions Our aim is to describe the manifold structure on classical linear groups and from there deduce a number of results. Before we begin we make

More information

Topological properties

Topological properties CHAPTER 4 Topological properties 1. Connectedness Definitions and examples Basic properties Connected components Connected versus path connected, again 2. Compactness Definition and first examples Topological

More information

Course Summary Math 211

Course Summary Math 211 Course Summary Math 211 table of contents I. Functions of several variables. II. R n. III. Derivatives. IV. Taylor s Theorem. V. Differential Geometry. VI. Applications. 1. Best affine approximations.

More information

Lecture 2: Review of Prerequisites. Table of contents

Lecture 2: Review of Prerequisites. Table of contents Math 348 Fall 217 Lecture 2: Review of Prerequisites Disclaimer. As we have a textbook, this lecture note is for guidance and supplement only. It should not be relied on when preparing for exams. In this

More information

DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17

DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,

More information

Symplectic and Poisson Manifolds

Symplectic and Poisson Manifolds Symplectic and Poisson Manifolds Harry Smith In this survey we look at the basic definitions relating to symplectic manifolds and Poisson manifolds and consider different examples of these. We go on to

More information

Lecture 8. Connections

Lecture 8. Connections Lecture 8. Connections This lecture introduces connections, which are the machinery required to allow differentiation of vector fields. 8.1 Differentiating vector fields. The idea of differentiating vector

More information

INTRODUCTION TO LIE ALGEBRAS. LECTURE 1.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 1. INTRODUCTION TO LIE ALGEBRAS. LECTURE 1. 1. Algebras. Derivations. Definition of Lie algebra 1.1. Algebras. Let k be a field. An algebra over k (or k-algebra) is a vector space A endowed with a bilinear

More information

Linear Algebra March 16, 2019

Linear Algebra March 16, 2019 Linear Algebra March 16, 2019 2 Contents 0.1 Notation................................ 4 1 Systems of linear equations, and matrices 5 1.1 Systems of linear equations..................... 5 1.2 Augmented

More information

A brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström

A brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal

More information

Chapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves

Chapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction

More information

LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups

LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are

More information

z x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables.

z x = f x (x, y, a, b), z y = f y (x, y, a, b). F(x, y, z, z x, z y ) = 0. This is a PDE for the unknown function of two independent variables. Chapter 2 First order PDE 2.1 How and Why First order PDE appear? 2.1.1 Physical origins Conservation laws form one of the two fundamental parts of any mathematical model of Continuum Mechanics. These

More information

INTRODUCTION TO ALGEBRAIC GEOMETRY

INTRODUCTION TO ALGEBRAIC GEOMETRY INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1 Preliminary of Calculus on Manifolds 11 Tangent Vectors What are tangent vectors we encounter in Calculus? (1) Given a parametrised curve α(t) = ( x(t),

More information

LECTURE 5: SMOOTH MAPS. 1. Smooth Maps

LECTURE 5: SMOOTH MAPS. 1. Smooth Maps LECTURE 5: SMOOTH MAPS 1. Smooth Maps Recall that a smooth function on a smooth manifold M is a function f : M R so that for any chart 1 {ϕ α, U α, V α } of M, the function f ϕ 1 α is a smooth function

More information

Symmetric Spaces Toolkit

Symmetric Spaces Toolkit Symmetric Spaces Toolkit SFB/TR12 Langeoog, Nov. 1st 7th 2007 H. Sebert, S. Mandt Contents 1 Lie Groups and Lie Algebras 2 1.1 Matrix Lie Groups........................ 2 1.2 Lie Group Homomorphisms...................

More information

TANGENT VECTORS. THREE OR FOUR DEFINITIONS.

TANGENT VECTORS. THREE OR FOUR DEFINITIONS. TANGENT VECTORS. THREE OR FOUR DEFINITIONS. RMONT We define and try to understand the tangent space of a manifold Q at a point q, as well as vector fields on a manifold. The tangent space at q Q is a real

More information

Derivations and differentials

Derivations and differentials Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,

More information

The prototypes of smooth manifolds

The prototypes of smooth manifolds The prototypes of smooth manifolds The prototype smooth manifolds are the open subsets of R n. If U is an open subset of R n, a smooth map from U to R m is an m-tuple of real valued functions (f 1, f 2,...,

More information

Chap. 1. Some Differential Geometric Tools

Chap. 1. Some Differential Geometric Tools Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U

More information

Choice of Riemannian Metrics for Rigid Body Kinematics

Choice of Riemannian Metrics for Rigid Body Kinematics Choice of Riemannian Metrics for Rigid Body Kinematics Miloš Žefran1, Vijay Kumar 1 and Christopher Croke 2 1 General Robotics and Active Sensory Perception (GRASP) Laboratory 2 Department of Mathematics

More information

INVERSE FUNCTION THEOREM and SURFACES IN R n

INVERSE FUNCTION THEOREM and SURFACES IN R n INVERSE FUNCTION THEOREM and SURFACES IN R n Let f C k (U; R n ), with U R n open. Assume df(a) GL(R n ), where a U. The Inverse Function Theorem says there is an open neighborhood V U of a in R n so that

More information

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2.

INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More examples. Ideals. Direct products. 2.1. More examples. 2.1.1. Let k = R, L = R 3. Define [x, y] = x y the cross-product. Recall that the latter is defined

More information

Basic Concepts of Group Theory

Basic Concepts of Group Theory Chapter 1 Basic Concepts of Group Theory The theory of groups and vector spaces has many important applications in a number of branches of modern theoretical physics. These include the formal theory of

More information

Differentiation. f(x + h) f(x) Lh = L.

Differentiation. f(x + h) f(x) Lh = L. Analysis in R n Math 204, Section 30 Winter Quarter 2008 Paul Sally, e-mail: sally@math.uchicago.edu John Boller, e-mail: boller@math.uchicago.edu website: http://www.math.uchicago.edu/ boller/m203 Differentiation

More information

The goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T

The goal of this chapter is to study linear systems of ordinary differential equations: dt,..., dx ) T 1 1 Linear Systems The goal of this chapter is to study linear systems of ordinary differential equations: ẋ = Ax, x(0) = x 0, (1) where x R n, A is an n n matrix and ẋ = dx ( dt = dx1 dt,..., dx ) T n.

More information

Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds

Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds MA 755 Fall 05. Notes #1. I. Kogan. Manifolds and tangent bundles. Vector fields and flows. 1 Differential manifolds, smooth maps, submanifolds Definition 1 An n-dimensional C k -differentiable manifold

More information

COMPLETE METRIC SPACES AND THE CONTRACTION MAPPING THEOREM

COMPLETE METRIC SPACES AND THE CONTRACTION MAPPING THEOREM COMPLETE METRIC SPACES AND THE CONTRACTION MAPPING THEOREM A metric space (M, d) is a set M with a metric d(x, y), x, y M that has the properties d(x, y) = d(y, x), x, y M d(x, y) d(x, z) + d(z, y), x,

More information

Let us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group.

Let us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group. Chapter 1 SOME MATHEMATICAL TOOLS 1.1 Some definitions in algebra Let us recall in a nutshell the definition of some important algebraic structure, increasingly more refined than that of group. Ring A

More information

CHAPTER 1 PRELIMINARIES

CHAPTER 1 PRELIMINARIES CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable

More information

Appendix E : Note on regular curves in Euclidean spaces

Appendix E : Note on regular curves in Euclidean spaces Appendix E : Note on regular curves in Euclidean spaces In Section III.5 of the course notes we posed the following question: Suppose that U is a connected open subset of R n and x, y U. Is there a continuous

More information

g(t) = f(x 1 (t),..., x n (t)).

g(t) = f(x 1 (t),..., x n (t)). Reading: [Simon] p. 313-333, 833-836. 0.1 The Chain Rule Partial derivatives describe how a function changes in directions parallel to the coordinate axes. Now we shall demonstrate how the partial derivatives

More information

ALGEBRAIC GROUPS: PART III

ALGEBRAIC GROUPS: PART III ALGEBRAIC GROUPS: PART III EYAL Z. GOREN, MCGILL UNIVERSITY Contents 10. The Lie algebra of an algebraic group 47 10.1. Derivations 47 10.2. The tangent space 47 10.2.1. An intrinsic algebraic definition

More information

BROUWER FIXED POINT THEOREM. Contents 1. Introduction 1 2. Preliminaries 1 3. Brouwer fixed point theorem 3 Acknowledgments 8 References 8

BROUWER FIXED POINT THEOREM. Contents 1. Introduction 1 2. Preliminaries 1 3. Brouwer fixed point theorem 3 Acknowledgments 8 References 8 BROUWER FIXED POINT THEOREM DANIELE CARATELLI Abstract. This paper aims at proving the Brouwer fixed point theorem for smooth maps. The theorem states that any continuous (smooth in our proof) function

More information

DIFFERENTIAL GEOMETRY. LECTURE 12-13,

DIFFERENTIAL GEOMETRY. LECTURE 12-13, DIFFERENTIAL GEOMETRY. LECTURE 12-13, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of

More information

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C)

LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) LECTURE 3: REPRESENTATION THEORY OF SL 2 (C) AND sl 2 (C) IVAN LOSEV Introduction We proceed to studying the representation theory of algebraic groups and Lie algebras. Algebraic groups are the groups

More information

Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur

Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Modern Algebra Prof. Manindra Agrawal Department of Computer Science and Engineering Indian Institute of Technology, Kanpur Lecture 02 Groups: Subgroups and homomorphism (Refer Slide Time: 00:13) We looked

More information

3 Applications of partial differentiation

3 Applications of partial differentiation Advanced Calculus Chapter 3 Applications of partial differentiation 37 3 Applications of partial differentiation 3.1 Stationary points Higher derivatives Let U R 2 and f : U R. The partial derivatives

More information

STOKES THEOREM ON MANIFOLDS

STOKES THEOREM ON MANIFOLDS STOKES THEOREM ON MANIFOLDS GIDEON DRESDNER Abstract. The generalization of the Fundamental Theorem of Calculus to higher dimensions requires fairly sophisticated geometric and algebraic machinery. In

More information

Math 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech

Math 6455 Nov 1, Differential Geometry I Fall 2006, Georgia Tech Math 6455 Nov 1, 26 1 Differential Geometry I Fall 26, Georgia Tech Lecture Notes 14 Connections Suppose that we have a vector field X on a Riemannian manifold M. How can we measure how much X is changing

More information

These notes are incomplete they will be updated regularly.

These notes are incomplete they will be updated regularly. These notes are incomplete they will be updated regularly. LIE GROUPS, LIE ALGEBRAS, AND REPRESENTATIONS SPRING SEMESTER 2008 RICHARD A. WENTWORTH Contents 1. Lie groups and Lie algebras 2 1.1. Definition

More information

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM

LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM LECTURE 10: THE ATIYAH-GUILLEMIN-STERNBERG CONVEXITY THEOREM Contents 1. The Atiyah-Guillemin-Sternberg Convexity Theorem 1 2. Proof of the Atiyah-Guillemin-Sternberg Convexity theorem 3 3. Morse theory

More information

Abstract & Applied Linear Algebra (Chapters 1-2) James A. Bernhard University of Puget Sound

Abstract & Applied Linear Algebra (Chapters 1-2) James A. Bernhard University of Puget Sound Abstract & Applied Linear Algebra (Chapters 1-2) James A. Bernhard University of Puget Sound Copyright 2018 by James A. Bernhard Contents 1 Vector spaces 3 1.1 Definitions and basic properties.................

More information

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space

Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................

More information

D-MATH Alessio Savini. Exercise Sheet 4

D-MATH Alessio Savini. Exercise Sheet 4 ETH Zürich Exercise Sheet 4 Exercise 1 Let G, H be two Lie groups and let ϕ : G H be a smooth homomorphism. Show that ϕ has constant rank. Solution: We need to show that for every g G the rank of the linear

More information

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III

Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Representations of algebraic groups and their Lie algebras Jens Carsten Jantzen Lecture III Lie algebras. Let K be again an algebraically closed field. For the moment let G be an arbitrary algebraic group

More information

Chapter 3a Topics in differentiation. Problems in differentiation. Problems in differentiation. LC Abueg: mathematical economics

Chapter 3a Topics in differentiation. Problems in differentiation. Problems in differentiation. LC Abueg: mathematical economics Chapter 3a Topics in differentiation Lectures in Mathematical Economics L Cagandahan Abueg De La Salle University School of Economics Problems in differentiation Problems in differentiation Problem 1.

More information

NOTES ON DIFFERENTIAL FORMS. PART 1: FORMS ON R n

NOTES ON DIFFERENTIAL FORMS. PART 1: FORMS ON R n NOTES ON DIFFERENTIAL FORMS. PART 1: FORMS ON R n 1. What is a form? Since we re not following the development in Guillemin and Pollack, I d better write up an alternate approach. In this approach, we

More information

Formal Groups. Niki Myrto Mavraki

Formal Groups. Niki Myrto Mavraki Formal Groups Niki Myrto Mavraki Contents 1. Introduction 1 2. Some preliminaries 2 3. Formal Groups (1 dimensional) 2 4. Groups associated to formal groups 9 5. The Invariant Differential 11 6. The Formal

More information

be any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism

be any ring homomorphism and let s S be any element of S. Then there is a unique ring homomorphism 21. Polynomial rings Let us now turn out attention to determining the prime elements of a polynomial ring, where the coefficient ring is a field. We already know that such a polynomial ring is a UFD. Therefore

More information

THE EULER CHARACTERISTIC OF A LIE GROUP

THE EULER CHARACTERISTIC OF A LIE GROUP THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth

More information

Linear Algebra. Preliminary Lecture Notes

Linear Algebra. Preliminary Lecture Notes Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date April 29, 23 2 Contents Motivation for the course 5 2 Euclidean n dimensional Space 7 2. Definition of n Dimensional Euclidean Space...........

More information

with a given direct sum decomposition into even and odd pieces, and a map which is bilinear, satisfies the associative law for multiplication, and

with a given direct sum decomposition into even and odd pieces, and a map which is bilinear, satisfies the associative law for multiplication, and Chapter 2 Rules of calculus. 2.1 Superalgebras. A (commutative associative) superalgebra is a vector space A = A even A odd with a given direct sum decomposition into even and odd pieces, and a map A A

More information

Integration and Manifolds

Integration and Manifolds Integration and Manifolds Course No. 100 311 Fall 2007 Michael Stoll Contents 1. Manifolds 2 2. Differentiable Maps and Tangent Spaces 8 3. Vector Bundles and the Tangent Bundle 13 4. Orientation and Orientability

More information

Implicit Functions, Curves and Surfaces

Implicit Functions, Curves and Surfaces Chapter 11 Implicit Functions, Curves and Surfaces 11.1 Implicit Function Theorem Motivation. In many problems, objects or quantities of interest can only be described indirectly or implicitly. It is then

More information

As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).

As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted

More information

Review of Multi-Calculus (Study Guide for Spivak s CHAPTER ONE TO THREE)

Review of Multi-Calculus (Study Guide for Spivak s CHAPTER ONE TO THREE) Review of Multi-Calculus (Study Guide for Spivak s CHPTER ONE TO THREE) This material is for June 9 to 16 (Monday to Monday) Chapter I: Functions on R n Dot product and norm for vectors in R n : Let X

More information

Linear Algebra. Preliminary Lecture Notes

Linear Algebra. Preliminary Lecture Notes Linear Algebra Preliminary Lecture Notes Adolfo J. Rumbos c Draft date May 9, 29 2 Contents 1 Motivation for the course 5 2 Euclidean n dimensional Space 7 2.1 Definition of n Dimensional Euclidean Space...........

More information

Problems in Linear Algebra and Representation Theory

Problems in Linear Algebra and Representation Theory Problems in Linear Algebra and Representation Theory (Most of these were provided by Victor Ginzburg) The problems appearing below have varying level of difficulty. They are not listed in any specific

More information

i = f iα : φ i (U i ) ψ α (V α ) which satisfy 1 ) Df iα = Df jβ D(φ j φ 1 i ). (39)

i = f iα : φ i (U i ) ψ α (V α ) which satisfy 1 ) Df iα = Df jβ D(φ j φ 1 i ). (39) 2.3 The derivative A description of the tangent bundle is not complete without defining the derivative of a general smooth map of manifolds f : M N. Such a map may be defined locally in charts (U i, φ

More information

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, )

Eilenberg-Steenrod properties. (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, ) II.3 : Eilenberg-Steenrod properties (Hatcher, 2.1, 2.3, 3.1; Conlon, 2.6, 8.1, 8.3 8.5 Definition. Let U be an open subset of R n for some n. The de Rham cohomology groups (U are the cohomology groups

More information

Or, more succinctly, lim

Or, more succinctly, lim Lecture 7. Functions and Stuff PCMI Summer 2015 Undergraduate Lectures on Flag Varieties Lecture 7. Functions and differentiable and analytic manifolds. The implicit function theorem, cotangent spaces

More information

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition)

Vector Space Basics. 1 Abstract Vector Spaces. 1. (commutativity of vector addition) u + v = v + u. 2. (associativity of vector addition) Vector Space Basics (Remark: these notes are highly formal and may be a useful reference to some students however I am also posting Ray Heitmann's notes to Canvas for students interested in a direct computational

More information

CHAPTER 7. Connectedness

CHAPTER 7. Connectedness CHAPTER 7 Connectedness 7.1. Connected topological spaces Definition 7.1. A topological space (X, T X ) is said to be connected if there is no continuous surjection f : X {0, 1} where the two point set

More information

Matrix Algebra: Vectors

Matrix Algebra: Vectors A Matrix Algebra: Vectors A Appendix A: MATRIX ALGEBRA: VECTORS A 2 A MOTIVATION Matrix notation was invented primarily to express linear algebra relations in compact form Compactness enhances visualization

More information

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori

LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI. 1. Maximal Tori LECTURE 25-26: CARTAN S THEOREM OF MAXIMAL TORI 1. Maximal Tori By a torus we mean a compact connected abelian Lie group, so a torus is a Lie group that is isomorphic to T n = R n /Z n. Definition 1.1.

More information

Holomorphic line bundles

Holomorphic line bundles Chapter 2 Holomorphic line bundles In the absence of non-constant holomorphic functions X! C on a compact complex manifold, we turn to the next best thing, holomorphic sections of line bundles (i.e., rank

More information

LECTURE 10: THE PARALLEL TRANSPORT

LECTURE 10: THE PARALLEL TRANSPORT LECTURE 10: THE PARALLEL TRANSPORT 1. The parallel transport We shall start with the geometric meaning of linear connections. Suppose M is a smooth manifold with a linear connection. Let γ : [a, b] M be

More information

1 Categorical Background

1 Categorical Background 1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,

More information

1 Functions of many variables.

1 Functions of many variables. MA213 Sathaye Notes on Multivariate Functions. 1 Functions of many variables. 1.1 Plotting. We consider functions like z = f(x, y). Unlike functions of one variable, the graph of such a function has to

More information

2. Intersection Multiplicities

2. Intersection Multiplicities 2. Intersection Multiplicities 11 2. Intersection Multiplicities Let us start our study of curves by introducing the concept of intersection multiplicity, which will be central throughout these notes.

More information

u n 2 4 u n 36 u n 1, n 1.

u n 2 4 u n 36 u n 1, n 1. Exercise 1 Let (u n ) be the sequence defined by Set v n = u n 1 x+ u n and f (x) = 4 x. 1. Solve the equations f (x) = 1 and f (x) =. u 0 = 0, n Z +, u n+1 = u n + 4 u n.. Prove that if u n < 1, then

More information

Lecture 2: Controllability of nonlinear systems

Lecture 2: Controllability of nonlinear systems DISC Systems and Control Theory of Nonlinear Systems 1 Lecture 2: Controllability of nonlinear systems Nonlinear Dynamical Control Systems, Chapter 3 See www.math.rug.nl/ arjan (under teaching) for info

More information