Introductory Riemannian Geometry 1: Differential Geometry Primer

Topic
Differential Geometry foundations for Riemmanian geometry. Part of the series outlined at Introductory Riemannian Geometry

Seminar
 16:54:14 Jafet: kommodore, what kind of prerequisites for today's audience do you have in mind? 16:56:48 kommodore: Differentiation in R^n, I'll try to start by defining manifolds and move quickly to vector bundles

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17:00:08 kommodore: Shall I start? 17:00:38 kommodore: OK. Today I am supposed to talk about some differential geometry foundation to Riemannian geometry. This is going to be very boring for many, because it involves many definitions and I probably don't have enough time to give interesting examples. 17:01:03 kommodore: For those of you wanting interesting examples, maybe come back next week.... 17:01:10 kommodore: I assume everyone here knows what differentiation in R^n means. 17:01:33 kommodore: So let's get started with some historical remarks and motivations, as is the usual procedure for anyone giving a                   seminar. 17:01:41 kommodore: Riemannian geometry started with Riemann (who else?) in his 1854 thesis, in which he generalised some of Gauss' ideas to                   higher dimensions. It received very little attention back then, until Einstein came along and show it is a &quot;useful&quot; subject and deserves attention. 17:02:32 kommodore: With Weyl formalising the notion of manifolds in circa 1912, people started to generalise many theorems on surfaces to higher dimensional manifold. Many classical theorems in Riemannian geometry typicially have two or more names attached, one for the discoverer in 2-D, and one for n-D. 17:03:25 kommodore: To name two examples: Bonnet-Meyer (if Ricci&gt;k everywhere, then                   diam&lt;pi/sqrt(k)), Chern-Gauss-Bonnet (Euler characteristic=integral of                    some expression of curvature). Both of these I hope to cover later when I got time.... 17:03:59 kommodore: So let's start with the basic object of study --- manifolds. 17:04:25 kommodore: A (topological) manifold of dimension m is, roughly speaking, something which looks locally like R^m. The technical definition is: a Hausdorff, paracompact space locally homeomorphic to R^m. 17:05:07 kommodore: For this seminar, I will additionally assume connectedness, unless stated otherwise 17:05:47 kommodore: The Hausdorff and paracompact condition is just technicality that is                   used to rule out some exotic examples and give us a nice tool called the partition of unity, the locally homeomorphic to R^m condition is                   the really important condition here. 17:06:31 kommodore: We usually denote the manifold with uppercase alphabets such as M, and if we want to stress the dimension, we write it as M^m. 17:07:16 kommodore: To start doing calculus, we want a bit more smoothness, called the differential structure. So we define a smooth manifold to be a                   manifold with the additional property that, for any two overlapping local neighbourhoods phi: U-&gt;R^m, psi: V-&gt;R^m, the &quot;transition function&quot; is the composition 17:07:55 kommodore: psi o phi^{-1}: phi(U\cap V) (in R^m) --&gt; U\cap V --&gt; psi(U\cap V) (in                   R^m) 17:08:17 kommodore: As a map between open subsets of R^m, we want it to be infinitely differentiable. 17:08:54 kommodore: Here is a good place to put the usual abuse of notation note. We do                   not put explicit restriction of function's domain here (the \vert_{U\cap V}'s) because it just obscure what is really going on here. 17:09:40 kommodore: A collection (phi_i,U_i) with Union U_i=M is called an atlas for M.                   There is the usual convention that a manifold is equipped with its maximal atlas, unless otherwise specified. 17:10:16 kommodore: The maximality is not a big restriction at all, because once you have an atlas, there is a unique maximal atlas containing it, viz. throwing all charts that are smoothly compatible with your atlas into the maximal atlas. 17:10:40 ness: ! 17:10:48 kommodore: go ahead 17:11:05 ness: can there be distinct maximal atlases for the same manifold? 17:11:52 kommodore: yes for topological manifolds. For example, the topological 7-sphere has 28 distinct differential structures 17:12:29 ness: thanks 17:12:45 kommodore: those 27 that are different from the usual S^7 in R^8 are called exotic 17:13:24 kommodore: so... examples of manifolds 17:13:44 kommodore: R^n is obviously a manifold, so is any open subset of R^n 17:14:37 kommodore: As an example of that, the group of invertible matrices GL(n,R) is a                   manifold 17:15:29 kommodore: The submersion and immersion theorems are the basic high-brow tool for constructing a lot of smooth manifolds. 17:16:17 kommodore: we define a map f:M^m-&gt;N^n between smooth manifolds to be smooth if, for all p in M^m and every local neighbourhood phi: p in U in                   M^m -&gt; R^m, psi: f(p) in V in N^n -&gt; R^n, the map psi o f o phi^{-1} is smooth 17:17:09 kommodore: Now we will introduce two important object constructed from this manifold M, namely the tangent bundle and the cotangent bundle. 17:17:27 kommodore: Before that, we need to make precise our intuition about tangent vectors and differentials. 17:17:56 kommodore: There are at least 4 different ways to define what a tangent vector is. I'll just pick the most visual one: A tangent vector at a point p is                    an equivalence class of smooth curves through p.  The equivalence relation is the following: 17:18:35 kommodore: Since the point p is on M^m, there is a chart phi: p in U-&gt; R^m. Then two curves gamma, delta: (-epsilon,epsilon)-&gt;M^m with gamma(0)=delta(0)=p are equivalent if                   (phi o gamma)'(0)=(phi o delta)'(0). 17:19:02 kommodore: This definition is independent of the chart phi in the maximal atlas being used, because the chain rule tells us that switching from phi to                   some other psi just multiply both sides of the definition by the deriviative of (psi o phi^{-1}) at phi(p), which is an invertible linear map of R^m. 17:19:57 kommodore: The tangent vector corresponding to phi(p)+t*e_i, where e_i is the i-th basis vector for R^m, is usually denoted by @_i=@/@x^i (@=\partial). Note that this is honestly a derivation (it acts on                   functions by f|-&gt; @f/@x^i), and we can also now make sense of the familiar chain rule @/@y^i=(@x^j/@y^i)(@/@x^j). 17:21:03 kommodore: Note that Einstein summation convention is used here. In general, if                   an index appear both upstair and downstair once, we sum over that index. 17:21:59 kommodore: the usual vector space structure of R^m induces a vector space structure on tangent space at p, T_p(M)={tangent vectors at p to M}. 17:22:30 kommodore: Now we can define the tangent bundle TM. The tangent bundle is the collection of all tangent spaces. It is a smooth manifold when we give the obvious charts induced by the charts of M and R^m. Namely, 17:22:56 kommodore: if phi is a chart at p with neighbourhood U, then from the construction, we can pick representative for any tangent vector v to be                   the curve that is phi(p)+tv under phi. Since phi is smooth, this gives a chart from collection of all tangent vectors at some point of U, to                   phi(U)xR^m. 17:23:32 kommodore: The transition function is just (transition function for phi,psi on M,                   its derivative). So we do indeed get a smooth structure. 17:24:23 kommodore: The cotangent bundle T^*M is similarly constructed, using the cotangent space (dual to the tangent space). Elements of the cotangent bundle are called differentials. The dual to @_i is dx^i. 17:24:55 kommodore: Also, we can replace R^m by any vector space V, and we have a covering of M by charts phi_i, then for any collection of smoothly varying function (in p) phi_{ij}(p) in GL(V), subjected to three conditions: 17:25:13 kommodore: 1. phi_{ii}=id; 2. phi_{ij}=phi_{ji}^{-1}; 3. phi_{ij}phi_{jk}=phi_{ik} 17:26:11 kommodore: then we can repeat this construction to give a _vector bundle_ E-&gt;M, with fibre V 17:27:12 kommodore: i.e. the transition from phi_i to phi_j is via (phi_j o phi_i^{-1},                   phi_{ij}) 17:27:31 kommodore: So we can construct tensor powers of TM and T^*M. In particular, the smooth sections of the k-th exterior powers of T^*M are called differential k-forms on M, and is usually denoted by Omega^k(M). 17:28:12 kommodore: Example: Omega^0(M) is just the space of smooth functions on M. 17:29:34 kommodore: Vector bundles isomorphic to a product MxV are called trivial bundles 17:30:08 kommodore: not every vector bundle is trivial, for example, TS^2 is not trivial (this is the so-called hairy-ball theorem) 17:30:34 kommodore: A smooth f:M-&gt;N induces a map between tangent bundles, i.e.                   [gamma] -&gt; [f o gamma]. This map is well-defined, linear on each tangent space, and is usually denoted by df or f_*, called the differential or derivative of f. 17:32:31 kommodore: The dual map (f_*)^* is usually just denoted by f^*, a map from T^*N-&gt;T^*M 17:33:14 kommodore: The exterior derivative operator d:Omega^0(M)-&gt;Omega^1(M) can be                   generalised to taking Omega^k to Omega^(k+1). It is just 17:33:37 kommodore: f dx^{i_1}\wedge...\wedge dx^{i_k} -&gt; (df)\wedge dx^{i_1}\wedge...\wedge dx^{i_k} extended linearly 17:34:09 kommodore: In other words, we can write (d alpha)_{ij...k}=(@/@x^{[i})alpha_{j..k]}, where [...] is the antisymmetrisation of indices. 17:34:45 kommodore: If the manifold is oriented (i.e. you can choose all transition                   functions to have positive determinant), then we have a theory for integrating differential m-forms \int_M: Omega^m -&gt; R 17:35:07 kommodore: First we use partition of unity to restrict to integrating differential m-forms supported inside a coordinate neighbourhood. 17:35:36 kommodore: Then we define the integral of the differential m-form f(x) dx^1\wedge dx^2\wedge ... \wedge dx^m on R^m is just the usual integral \int f(x) dx^1...dx^m 17:35:58 kommodore: and transport back to the manifold. Later (i.e. next seminar if I get                   to it) we will see that this partition of unity is not necessary --- there is a closed set of dimension &lt;m which, when deleted from the manifold, gives us an open set diffeomorphic to R^m. 17:36:41 kommodore: The exterior derivative d satisfies d o d = 0, because of symmetry of                   partial derivatives. So (\Omega,d) is a complex, called the de Rham complex. The cohomology of this complex is the de Rham cohomology. 17:37:03 kommodore: A manifold-with-boundary is an obvious generalisation, making charts take value on the closed upper half space H^n instead of R^n. Then Stoke's theorem tells us that with respect to the pairing (m-manifold,                   m-forms)-&gt;R, d and @, the boundary operator, are adjoint. The proof of                   Stokes is by applying partition of unity, assume the form has compact support, then integrate just the first variable as in multivariate calculus. 17:37:49 kommodore: Finally, I get to talk about connexions.... 17:38:11 kommodore: Abstractly speaking, given a vector bundle E-&gt;M, there are no way of                   comparing adjacent vector spaces E_p and E_q for p,q in some trivialising neighbourhoods (which is what we need for &quot;directional                   derivative&quot;). This is because we can always twist any trivialisation with any smoothly varying GL(V)-valued function. 17:38:58 kommodore: So in order to compare, we need some way of saying what the &quot;constants&quot; are. 17:39:11 kommodore: for any p in M, we have an obvious exact sequence 17:39:51 kommodore: 0 -&gt; V -&gt; T_pE -&gt; T_{pi(p)}M -&gt; 0; pi:E-&gt;M 17:40:29 kommodore: oops, make that p in E 17:40:54 kommodore: We want to split this exact sequence 17:41:27 kommodore: The image of T_{pi(p)}M-&gt;T_pE is called the horizontal subspace 17:41:45 kommodore: and image of V-&gt;T_pE is the vertical subspace 17:42:00 kommodore: Note that V=ker(d pi), so the vertical subspace is independent of                   trivialisation. 17:42:25 kommodore: So if we have a splitting, then there must be (dim V) linearly independent linear functionals theta^1,...,theta^{dim V} such that T_pM appears in TE as the common kernel. Now if we write a^i for the V-coordinates and x^k for the M-coordinates in some trivialisation, then we can choose 17:42:49 kommodore: theta^i=da^i+e^i_k(a,x) dx^k 17:43:11 kommodore: Moreover, if we further demand that e^j_k is _linear_ in a, i.e. 17:43:26 kommodore: e^i_k(a,x)=Gamma^i_{jk} a^j 17:43:51 kommodore: Then the horizontal subspace ker(theta^1,...,theta^{dim V}) is actually a linear subspace. The Gamma^i_{jk} are called the coefficients of the connexion. 17:44:38 kommodore: Note the connexion is not in general a differential form, but the difference between two connexions is a matrix of differential forms (we will see that later). So the space of connexion is an affine linear space. 17:45:02 kommodore: A connexion give rise to covariant derivative, as follows: 17:45:18 kommodore: A covariant derivative is a mapping D:(vector field)x(section of E)-&gt;(section of E) satisfying Omega^0(M)-linearity on the first coordinate, and behaves like a                   derivation on the second, i.e. R-linear with 17:45:37 kommodore: D(v,fs)= v(f)s + f D(v,s). 17:46:00 kommodore: If we move the vector field to the other side of the arrow, then D                   becomes a mapping (section of E)-&gt;(section of T^*M\otimes E). This is                   usually how one defines a covariant derivative. 17:46:38 kommodore: Then a connexion corresponds to a covariant derivative by: Gamma^i_{jk}=(e_i) component of D(@_k,e_j), where (e^i) is a basis for V, and extend by linearity. We also write D_v(s) for D(v,s). 17:47:02 kommodore: Given connexions on E-&gt;M, F-&gt;M, there are natural induced connexion of                   E\otimes F, E^*, tensor powers, etc, by requiring appropriate Leibniz rule to hold. 17:47:35 kommodore: We sometimes say &quot;a connexion on M&quot; for &quot;a connexion of TM-&gt;M&quot;. A connexion on M is torsion-free if Gamma^i_{jk} is symmetric in j and k. 17:48:36 kommodore: So now we can do first covariant derivative. What about higher ones? 17:48:59 kommodore: If we think of covariant derivative as a map from sections of E to                   sections of T^*M\otimes E, or equivalently, an End(E)-valued 1-forms of M (written Omega^1(M;End(E))), then we can write 17:49:27 kommodore: D=d+A\wedge, where A=(Gamma^i_{jk}dx^k) is a matrix of 1-forms of M. 17:49:53 kommodore: (here is where you see the difference of two connexions is a matrix of                   1-forms) 17:50:14 kommodore: Then we can repeat this procedure and have second, third, ...                   covariant derivatives. The second covariant derivative is 17:50:27 kommodore: DDs=d(ds+As)+A\wedge(ds+As) =(dA)s-A\wedge ds+A\wedge ds+(A\wedge A\wedge)s 17:50:50 kommodore: the minus sign comes from the fact that the entries of A are 1-forms, so the antisymmetrisation is going to give a minus sign 17:51:22 kommodore: so DD: s|-&gt;(dA+A\wedge A)s is actually a linear algebraic operator. We call F(A)=dA+A\wedge A the _curvature form_ of the connexion. If F(A)=0, we say the connexion is _flat_. 17:51:50 kommodore: The Bianchi identity for F(A) states that the curvature form is                   covariantly constant: D(F(A))=0, for any A defining a covariant derivative D. 17:52:16 kommodore: One-line Proof: This comes from the obvious (DD)D=D(DD) and Leibniz rule. QED. 17:52:52 kommodore: Example: the trivial connexion on the trivial bundle M\times V is                   D(u,f^iv_i)=u(f^i)v_i; i.e. just the usual derivative on each component. It is obviously flat 17:54:33 kommodore: Let me end today by introducing the Riemannian metric and cometrics, and define the Levi-Civita connexion 17:54:53 kommodore: A (smooth) Riemannian metric g on M is a smoothly varying positive definite symmetric bilinear form 17:55:07 kommodore: g_p: T_pM\times T_p M -&gt; R 17:55:25 kommodore: and we say (M,g) is a Riemannian manifold 17:55:56 kommodore: In local coordinates, we can write g=g_{ij}dx^idx^j 17:56:15 kommodore: Every manifold admit Riemannian metric, because the positive-definite condition is convex. So just saying a Riemannian manifold doesn't                   really tell you much about the manifold. (Contrast this with Lorenzian                   metric in GR --- AFAIK we don't have a full characterisation of                    manifolds admitting Lorenzian metrics) 17:56:55 kommodore: From linear algebra, we know g also induces a cometric g^*: T^*M\times T^*M -&gt; R. In local coordinates, g^*=g^{ij}@_i@_j, where (g^{ij}) is the inverse matrix to (g_{ij}). 17:57:34 kommodore: The Levi-Civita connexion is a connexion on M which is uniquely determined by two conditions: 17:57:43 kommodore: (1) D is torsion-free; and 17:58:11 kommodore: (2) g is D-covariantly constant, i.e.                        g(D_X(Y),Z)+g(Y,D_X(Z))=X(g(Y,Z)). 17:58:51 kommodore: The existence and uniqueness is what is known as the fundamental theorem of Riemannian geometry. We also have the Kozsul formula for D: 17:59:04 kommodore: 2g(D_X(Y),Z)=X(g(Y,Z))+Y(g(X,Z))-Z(g(X,Y)) + g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X) 17:59:38 kommodore: Example: if G is a compact Lie group with its bi-invariant metric, then the Levi-Civita connexion is D_X(Y)=[X,Y]/2 for left-invariant vector fields X,Y 18:00:09 kommodore: Proof: Let X,Y,Z be left-invariant vector fields, then the Kozsul formula gives 18:00:22 kommodore: 2g(D_X(Y),Z)=X(g(Y,Z))+Y(g(X,Z))-Z(g(X,Y)) + g([X,Y],Z)-g([X,Z],Y)-g([Y,Z],X) 18:00:43 kommodore: By bi-invariance, the first three terms cancel (because                   g(X,Y)=constant, etc.) 18:00:53 kommodore: and so we are left to prove the last two terms does not contribute. 18:01:23 kommodore: This is done by noticing the adjoint representation Ad is an isometry of the Lie algebra = T_{id}G 18:01:34 kommodore: (because it is a composition of left and right translation, so                   isometry by bi-invariance of metric) 18:01:42 kommodore: so 18:01:52 kommodore: (d/dt|_{t=0})g(Ad_(exp(tZ))X, Ad_{exp(tZ)}Y)=0 18:02:10 kommodore: so upon remembering the derivative of Ad is ad, which is the Lie bracket, we get the last two terms cancel. QED. 18:02:37 kommodore: Questions? Maybe I've bored everyone out of existence? 18:04:27 _llll_: i was lost after about 20mins, i dindt really understand what TM was, or                how it was a manifold 18:05:04 ness: me too 18:05:55 kommodore: when M=open set of R^m, TM=MxR^m 18:06:07 kommodore: so that is a manifold 18:06:40 _llll_: probably me being slow, but what *is* TM? 18:06:42 kommodore: you can visualise this TM as the space of arrows having base at some point of M 18:07:21 kommodore: TM=Union_{p in M} T_pM as a set 18:07:41 ness: how can v \in TM &quot;act&quot; on elements of M (or functions on M?)? 18:08:11 kommodore: essentially this is done by taking directional derivative 18:08:15 ness: I didn't at all get the part where v is a kind of derivative 18:08:35 kommodore: my definition of v is an equivalence class of curves, right? 18:08:40 ness: yes 18:09:20 kommodore: so you can make sense of (f o gamma)(t) for t in (-epsilon,epsilon), where gamma represents v 18:09:49 ness: ok 18:09:53 kommodore: then we define v(f) to be (f o gamma)'(0) 18:10:01 ness: oh 18:10:14 _llll_: ah... makes a bit more sense 18:10:40 kommodore: yes 18:11:09 kommodore: there are other equivalent definitions of what a tangent vector is 18:11:22 kommodore: one of them is a derivation at p.... 18:12:20 ness: kommodore: would you mind me asking more basic questions like this? 18:12:24 kommodore: another one is &quot;something whose components transform according to                   @/@y^i=(@x^j/@y^i)(@/@x^j) 18:12:39 kommodore: ness: not at all 18:13:47 ness: what *is* the cotangent bundle T^*M? or to start with, what is T_p^*M? Is that just the dual space of T_pM 18:13:54 kommodore: ness: yes 18:14:58 eigenval: some year ago i tried to understand general relativity. i had not. but i've remebered something, here. nice compendium of the underlying maths :-). my question: what is an interpretation of the                  torsion-freeness of a connexion? 18:16:19 ness: So T_p^*M is the space if linear functionals from directional derivatives               at p to R. In what sense are elements of T_p^*M (they are called differentials, right?) related to &quot;traditional&quot; differentials             (I do understand that traditional differentials are't so well defined at               all)? 18:18:21 kommodore: eigenval: connexions with torsions are generally a pain to                    work with... you don't have the Bianchi identities 18:19:26 kommodore: ness: the traditional differentials is supposedly transform in the                    same way as elements of T_p^*M 18:19:57 kommodore: i.e. you want df=(df/dg)dg in 1-dimension 18:21:19 kommodore: and the obvious higher-dimensional analogue df=(@f/@x^i)dx^i 18:22:47 ness: here df and dg are elements of T_p^*M, and (df/dg) is normal derivation? 18:23:40 kommodore: df and dg are &quot;traditional differentials&quot;/&quot;elements of T_p^*M&quot;,                    the same formula holds 18:24:50 kommodore: and using these, we have apparently &quot;proved&quot; the change of variable formula in multiple integrals.... 18:26:07 ness: indeed 18:28:10 kommodore: because dx^1\wedge...\wedge dx^n =[(@x^1/@y^{i_1})dy^{i_1}]\wedge ...\wedge [(@x^n/@y^{i_n})dy^{i_n}] = ...                      =\sum_{sigma in S_n} (@x^1/@y^{sigma(1)})...(@x^n/@y^{sigma(n)}) sign(sigma) dy^1...dy^n 18:30:17 kommodore:   = J(x;y) dy^1\wedge...\wedge dy^n 18:30:36 ness: which is an expression for the jacobian determinant. and where the sign changes follow from the antsymmetry properties, right? 18:30:44 kommodore: yes 18:30:53 kommodore: so now stick the integral sign and voila! 18:31:26 ness: after defining the cotangent bundle you go on to &quot;replace R^n by any vector space V&quot; and you sorta completely lost me here. Can you describe what this is about? are you &quot;generalizing&quot; manifolds to look locally like V instead of 18:31:34 ness: R^n 18:32:04 kommodore: R^m was referring to the tangent/cotangent space 18:33:22 kommodore: so now instead of forcing how we patch the U_ixR^m together (by (phi_j o phi_i^{-1}, Derivative of that)), we now want to stick U_ixV's together, where V is some vector space 18:35:44 kommodore: the thing that we replace is R^m by V, and &quot;Derivative of                   (phi_j o phi_i^{-1})&quot; by some smoothly varying GL(V)-valued function from U_i\cap U_j 18:36:34 kommodore: that GL(V)-valued function from U_i\cap U_j I denote by phi_{ij} 18:36:42 _llll_: i think you're just defining &quot;vector bundle&quot; here? 18:36:50 kommodore: yes 18:38:20 kommodore: so we are doing E=(Union_i U_ixV)/{identifying (u,v) in U_ixV with (u,phi_{ij}(v)) in U_jxV for u in U_i\cap U_j} 18:40:08 kommodore: I didn't mention it in the talk, but GL(V) can be replaced by any subgoup of GL(V) 18:40:24 _llll_: makes much more sense if you know about sheaves and etale spaces 18:41:09 kommodore: well, sheves is a generalisation of what we do with vector bundles... 18:42:19 _llll_: so the result here is that TM is a vector bundle over M i suppose, is it                also a manifold? 18:43:05 kommodore: yes, did I not say that in the talk?.... searching 18:44:27 _llll_: it may be in there, but if so, i didnt follow it :) 18:45:00 kommodore: I did... the obvious charts to cover TM are the U_ixR^n chart with                   transition function (U_i\cap U_j)xR^n being (phi_j o phi_i^{-1}, derivative of that) 18:45:55 _llll_: can you explain that a bit further? a chart for M is U--&gt;R^dim(M) ? 18:45:59 kommodore: err... not saying that coherently 18:46:50 kommodore: a chart for M is U-&gt;R^dim(M), giving rise to a chart of TM                   TU-&gt;R^dim(M)xR^dim(M) 18:47:25 kommodore: and TU may be identified with phi(U)xR^dim(M) 18:47:56 kommodore: because in R^dim(M) you have translating by (u_2-u_1) 18:48:22 kommodore: which maps (curves through u_1) to (curves through u_2) 18:48:32 _llll_: is this meant to be obvious how to construct TU-&gt;R^dim(M)xR^dim(M)                 from U--&gt;R^dim(M)? 18:50:00 kommodore: yes... you identify (u,v) in R^dim(M)xR^dim(M) with the curve                    t|-&gt;u+tv, which is an element of T_u(R^dim(M)) 18:51:20 kommodore: if U--&gt;R^dim(M), we might as well think of it as an open subset of                    R^dim(M), so this gives TU 18:53:00 _llll_: maybe it's just notation, or more likely it's me, but i dont find any of                 this clear so far 18:53:06 kommodore: I didn't dare to mention naturality and stuff like that, but the construction of all tensor bundles T^{(r,s)}M are natural 18:54:32 kommodore: Let's call it phi: U-&gt;R^dim(M) 18:54:54 kommodore: then we have constructed T(phi(U)), right? 18:55:30 _llll_: ok 18:56:35 kommodore: Now we say that *is* TU, by identifying u with phi(u) for all u in U,                   in the first R^dim(M) coordinate of T(phi(U)) 18:59:06 kommodore: so we get TU=UxR^dim(M), via this phi 19:02:13 kommodore: the problem is then: what happens if we have another chart V which overlaps with U? In the middle we get T(U\cap V) which we identify with an open subset of R^{2 dim(M)} in two different ways 19:03:21 eigenval: let me repeat, what i believe to have understood: we have the &quot;fundamental theorem of Riemannian geometry&quot;: there is a unique connextion, the L-C-connextion, such that the corresponding D                   is (2) compatible with the given riemann metric and that is                    (1) torsion-free. so why one takes that torsion-free connextion? just because one can easier work with it? 19:06:10 kommodore: eigenval: torsion-free connexions has many nice properties. If we dropped the torsion-free condition, there will be no uniqueness - just add any torsion 19:05:39 _llll_: are you just applying T to phi:U--&gt;R^m to get T(phi):TU-&gt;T(R^m)~R^m x R^m ? 19:06:44 kommodore: _llll_: yes. 19:06:53 _llll_: ah, ok i sort of follow a bit now 19:07:08 _llll_: so that makes TM a manifold, and also a vector bundle with fibre R^m 19:07:12 kommodore: The hard work then is to prove T is natural 19:08:11 kommodore: T(phi)=(d phi) 19:09:10 kommodore: proving T is natural is what the construction will show 19:13:20 _llll_: so presumably, if f:V--&gt;W then TV --&gt; VxR^m --fxid--&gt; WxR^m --&gt;TW is the map T(f)? 19:13:24 kommodore: To be honest, we really only need the tensor bundles only in what I planned, not the full-blown generality of vector bundles 19:13:44 kommodore: _llll_: no 19:14:00 _llll_: oh 19:14:19 kommodore: You will need to transform the second coordinate by the derivative of                   f at the relevant point 19:16:26 kommodore: so Tf(v,x)=(f(v), (df)(v)(x)), when V,W are open subsets of R^n, R^m 19:17:40 kommodore: because a curve x+tv is mapped under f to                   f(x+tv)=f(x)+tf'(x)(v)+higher order 19:18:11 _llll_: ah, interesting 19:22:11 _llll_: so could you have a higher order version of TM where the equivalence relation defining the tangent vectors involves the double derivative not just the single derivative at zero? 19:22:15 kommodore: The tangent bundle is a special case of the jet-bundle, which is                   keeping track of all Taylor coefficients. The tangent bundle keep only the first order information 19:23:12 kommodore: The transformation rule for jet bundles are horrible to write out 19:24:02 kommodore: If you keep up to the k-th order information of a map M-&gt;N, it is                   called the k-jet bundle J^k(M,N) 19:24:40 kommodore: (the J is probably going to be in calligraphic letter) 19:39:56 ~mary]]: well, transformation rules for jet bundles may be horrible to write out, but they aren't really that bad... if you look at the diagonal map d : M -&gt; M x M, the ideal sheaf defining the closed submanifold M is I.                 The cotangent bundle is basically I/I^2. Looking at I^n/I^{n+1} will get you higher order taylor coeff's

19:36:38 _llll_: what is the topic for next week? 19:40:17 kommodore: &quot;Riemannian geometry II: Curvature&quot; and I'll talk about the various curvature tensors, maybe onto how they affect the global topology

20:22:26 ChanServ changed the topic of #mathematics to: NEXT SEMINAR: Introductory Riemannian Geometry 2: Curvature by kommodore on Sunday 10 August 16:00UTC | Transcript of last seminar: http://www.freenode-math.com/Introductory_Riemannian_Geometry_1:_Differential_Geometry_Primer | Other seminars (past and future): http://www.freenode-math.com/index.php/Seminars