Christoffel Revisited Bas Fagginger Auer September 1, 2009 Master's Thesis Mathematical Institute Utrecht University Supervisor: Prof. Dr. J. J. Duistermaat This is an digitally edited photograph of Elwin Bruno Christoffel, the origi- nal of which was taken from the History of Mathematics archive of the School of Mathematics and Statistics of the University of St Andrews, Scotland, http://www-history.mcs.st-andrews.ac.uk/PictDisplay/Christoffel.html. Acknowledgements Before we start with the actual thesis, I would very much like to thank a few people whose input and putting up with my ramblings have made it possible to finish this thesis in its current form. I am very much indebted to my supervisor, professor Hans Duistermaat, for introducing me to this subject and helping me overcome many mathematical difficulties (in particular showing me simpler and more elegant ways to prove a great number of results) during our fruitful and pleasant discussions. My girlfriend, Hedwig van Driel, for painstakingly proofreading this entire document, her care, stuffing me with food, our lovely evenings, and being firm with me when I was procrastinating; thank you. Finally I would like to thank the numerous people whom I have bothered with my questions and unasked-for exposition of my results, in particular: Math- ijs Wintraecken, Matthijs van Dorp, Job Kuit, Albert-Jan Yzelman, Jan Jitse Venselaar, and Jaap Eldering, as well as my parents for their continued support and interest. Thank you all for your kind help, Bas Fagginger Auer. Abstract This thesis discusses E. B. Christoffel’s famous article from 1869 (an English translation is also included), generalises it to the more general setting of locally convex Hausdorff topological vector spaces over R or C, and furthermore estab- lishes a partial converse to the results of Christoffel in Banach spaces. Apart from this we rigorously discuss the aforementioned more general setting by in- troducing the concepts related to this setting from the ground up. This includes in particular a non-standard notion for the derivative and proofs of many useful statements, among which the open mapping theorem, closed graph theorem, fundamental theorem of integration, and the Taylor approximation theorem. Contents 1 Introduction3 1.1 Notation................................4 1.2 Overview...............................6 2 Topology8 2.1 Topological spaces..........................8 2.2 Separation axioms.......................... 18 2.3 Sequences............................... 21 2.4 Compactness............................. 23 2.5 Metric spaces............................. 25 3 Algebra 33 3.1 Groups................................. 33 3.2 Rings................................. 37 3.3 Modules................................ 39 4 Topology and algebra 50 4.1 Topological modules......................... 50 4.2 Normed modules........................... 54 4.3 Topological vector spaces...................... 56 4.4 F-spaces................................ 59 4.5 Local convexity............................ 65 5 Analysis 75 5.1 Differentiation............................ 75 5.2 Multilinear families.......................... 88 5.3 Integration.............................. 91 5.4 Fr´echet spaces............................. 105 5.5 Banach spaces............................. 108 - 1 CONTENTS 6 Revisiting Christoffel’s article 123 6.1 Preliminaries............................. 123 6.2 Generalisation............................. 125 6.3 Digression............................... 133 6.4 Simple metrics............................ 139 6.5 Making metrics simple........................ 146 6.6 Digression (cont'd).......................... 150 7 Conclusion 159 8 Translation 161 Bibliography 185 - 2 - Chapter 1 Introduction Welcome to this thesis, which is concerned with generalising E. B. Christoffel’s article, [Chr1869], and developing the underlying theory of the setting in which this generalisation should take place. The level of the material contained in this thesis should be appropriate for any master student of Mathematics, as we de- velop the theory mostly from the ground up, relying only on results established in basic set theory and analysis on R and C. However, a familiarity with topol- ogy, higher-dimensional analysis, and differential geometry will be very helpful for understanding the structure of this document, the included examples, and reasons for adopting certain definitions. We are interested in Christoffel’s article, because of its paramount impor- tance for the development of differential geometry at the end of the 19th century. Through the introduction of the Christoffel symbols (in [Chr1869] denoted by ij , but in differential geometry conventionally by Γ), the curvature tensor (in k [Chr1869] denoted by (ijkl) and now usually by R), and the means of covariant differentiation by using the Christoffel symbols, Christoffel provided very useful tools for the further development of differential geometry, as was undertaken by Gregorio Ricci-Curbastro and Tullio Levi-Civita (see [StAndrews]). These developments in turn permitted Albert Einstein to formulate his the- ory of general relativity entirely in terms of differential geometry, which was a major step in the physical modeling of the effects of gravity and electromag- netism in celestial mechanics. As differential geometry and general relativity are both still being practised by a great number of mathematicians and physicists today, this makes Christof- fel's article highly influential and very interesting to further investigate. Even more so because of the geometrical way in which the Christoffel symbols are currently introduced in differential geometry (via an affine connection on a vec- tor bundle, see [Ban2008]), which is not at all like the algebraic way in which they were used by Christoffel as tools to determine whether or not two given metrics could be transformed into one another via an appropriate coordinate transformation. - 3 1.1. NOTATION 1.1 Notation To ensure a concise treatment of the discussed material, we will strive to use the same symbols to denote the same type of objects. However, this is not always possible when a large number of objects is being discussed at the same time, so the following table is only meant to give an indication. A, B, . Sets. 0 a, a , a1, a2, . Elements of the set A. U, V , . Open subsets of A, B, . respectively. See Definition (2.1.2). A, B, . Collections of subsets of A, B, . respectively. f, g, . Functions between sets. i, j, . Indices of objects. k, l, . Elements of N. ∼ An equivalence relation. α, β, . Scalars, usually values in either R or C. For collections of numbers we will employ the usual notation. N The natural numbers: 1; 2; 3;:::. N0 The natural numbers together with zero: 0; 1; 2; 3;:::. N^ The natural numbers extended with infinity and considered as a topological space: 1; 2; 3;:::; 1. See Example (2.3.2). Z The integers: :::; −2; −1; 0; 1; 2; 3;:::. Q The rationals. R The real numbers. C The complex numbers, identified with the plane R2. K Refers to either R or C. ]α; β[ The open interval fγ 2 Rjα < γ < βg ⊆ R. [α; β] The closed interval fγ 2 Rjα ≤ γ ≤ βg ⊆ R. ]α; 1[ The open interval fγ 2 Rjα < γg ⊆ R. ] − 1; α[ The open interval fγ 2 Rjα > γg ⊆ R. ] − 1; 1[ The real line R. As well as the following notation for set operations. ; The empty set. A n B The complement of the set B in A, fa 2 Aja2 = Bg. S A The union of all sets in A, faj9A 2 A : a 2 Ag. S S Ai Defined as fAiji 2 Ig. i2I S A1 [ ::: [ Ak Defined as fA1;:::;Akg. T A The intersection of all sets in A, faj8A 2 A : a 2 Ag. T T Ai Defined as fAiji 2 Ig. i2I T A1 \ ::: \ Ak Defined as fA1;:::;Akg. ` A The disjoint union of all sets in A, the set f(A; a)jA 2 A; a 2 Ag. ` i2I Ai Defined as f(i; a)ji 2 I; a 2 Aig. Q A The product of all sets in A, the set fg : A! S Aj8A 2 A : g(A) 2 Ag. Q S i2I Ai Defined as fg : I ! i2I Aij8i 2 I : g(i) 2 Aig. A1 × ::: × Ak Defined as f(a1; : : : ; ak)ja1 2 A1; : : : ; ak 2 Akg. - 4 - 1.1. NOTATION Together with their usual identifications. 123 We will also use the following symbols. dom f The domain of a function, for f : A ! B, dom f := A. im f The image of a function, for f : A ! B, im f := f(A) := fb 2 Bj9a 2 A : f(a) = bg ⊆ B. graph The graph of a function f : A ! B, graph f := f(a; b) 2 A × Bjf(a) = bg ⊆ A × B. f −1(·) The pre-image of a set C ⊆ B under a function f : A ! B, f −1(C) := fa 2 Ajf(a) 2 Cg ⊆ A. idA The identity map, for a given set A, idA := A ! A : a 7! a. sgn The sign of a number, sgn : R ! {−1; 0; +1g where 8 < −1 α < 0 sgn(α) := 0 α = 0 : : +1 α > 0 Re, Im The real and imaginary parts of a complex number, for z = (x; y) = x + i y 2 C ' R2, Re(z) := x, Im(z) := y. P(A) The collection of subsets or powerset of a set A, P(A) := fBjB ⊆ Ag. int(A) Interior of a set A, Definition (2.1.2). A Closure of a set A, Definition (2.1.2). T (A) The topology generated by A, Definition (2.1.6). Sk Group of permutations of f1; : : : ; kg, Example (3.1.4). Abc Absorbent, balanced, and convex, Definition (4.3.4). k Daf The k-th derivative of a function f at a, Definition (5.1.1) and Definition (5.1.9). Ck(U; B) The set of all k-times continuously differentiable functions from an open set U ⊆ A to B, Definition (5.1.9). R β α f The integral of a function f over the interval [α; β], see Definition (5.3.2). L(A; B) Space of all continuous linear maps between Banach spaces A and B, see Definition (5.5.4).