contents.tex 34 KB
 Jan-Bernhard Kordaß committed Oct 25, 2016 1 2 3 \tableofcontents Jan-Bernhard Kordaß committed Oct 25, 2016 4 \chapter{Tosion Invariants [Roman Sauer]} Jan-Bernhard Kordaß committed Oct 25, 2016 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 Torsion invariants fall into a class of so-called secondary invariants'' of topological spaces in the sense that they are only defined if a certain class of primary invariants'' (e.g. Betti numbers) vanish. Often they reveal more subtle geometric information. The following will contain a discussion of Whitehead and Reidemeister torsion. Informally, corresponding primary invariants are Lefschetz numbers (Whitehead torsion) and the Euler characteristic (Reidemeister torsion). \section{Review of Euler characteristic and Lefschetz numbers.} \subsection{CW Complexes} \begin{dfn*} A (finite) \CmMark{CW-complex} is a hausdorff space with a decomposition $E$ into (finitely many) cells (space hemeomorphic to some $\R^n$) such that for every $e \in E$ there is a continuous map $\phi_e \colon D^n \to X$ with $\phi_e \colon \mathring D^n \xrightarrow{\cong} e$ and $\Im(\phi_e|_{S^{n-1}}) \subset \bigcup_{f \in E, \dim f \leq n-1} f$. \end{dfn*} \begin{expl*} \begin{enumerate} \item Simplicial complexes, e.g. triangles, pyramides, etc. \item But CW-complexes are more general, the following graph is CW for example: \begin{center} \begin{tikzpicture} \draw (0,0) to[bend left] (2,0); \draw (0,0) to[bend right] (2,0); \draw (2,0) to (3,0); \end{tikzpicture} \end{center} One can even attach a disc along its boundary to a single 1-cell. \end{enumerate} \end{expl*} \subsection{Euler characteristic} \begin{dfn*} The Euler class $\chi(X)$ of a finite CW-complex $X$ is defined as $\chi(X) = \sum_{i \geq 0}(-1)^i \#(i\text{-cells of } X) \in \Z$. \end{dfn*} \begin{thm*}[Euler-Poincaré] \begin{align*} \chi(X) = \sum_{i \geq 0} (-1)^i b_i(X), \end{align*} where $b_i(X) = \rk_{\Z} H_i(X;\Z)$. \end{thm*} In particular, $\chi$ is a homotopy invariant. \begin{proof}[Proof''] $H_i(X;\Z) = H_i(C_{*}^{CW}(X))$, where $C_{*}^{CW}(X)$ is the cellular chain complex \begin{align*} \cdot \to C_{i+1}^{CW}(X)\xrightarrow{\partial} \underbrace{C_{i}^{CW}(X)}_{\cong \Z^{\# i\text{-cells}}} \xrightarrow{\partial} C_{i-1}^{CW}(X) \to \cdots \end{align*} Thus $\chi(C_{*}) := \sum_{i \geq 0} (-1)^i\rk_{\Z}(C_{i})$ and $\chi(C^{CW}(X)) = \chi(X)$. This boils down to \begin{align*} \chi(C_{*}) = \sum_{i \geq 0} \rk_{\Z}H_i(C_{*}) ( = \chi(H_{*}(C_{*}))]. \end{align*} This is just additivity of the rank! Consider \begin{align*} C_1 \xrightarrow{\partial} C_0 \end{align*} and note that we have the exact sequences $0 \to \Im \partial \to C_0 \to \underbrace{H_0}_{= C_0/\Im \partial} \to 0$ and $0 \to \underbrace{H_1}_{= \Ker \partial} \to C_1 \xrightarrow{\partial} \Im \partial \to 0$. Thus $\chi(C_{*)} = \rk_{\Z} C_0 - \rk_{\Z} C_1 = \rk_{\Z} \Im \partial + \rk_{\Z} H_0 - \rk_{\Z}H_1 - \rk_{\Z} \Im \partial = \rk H_0 - \rk H_1$, which completes the proof''. \end{proof} \subsection{Review of cellular homology} Let $X$ be a CW-complex with cellular decomposition $E$. Then we can consider the \CmMark{n-skeleton} \begin{align*} X^n := \sum_{e \in E, \dim e \leq n} e, \end{align*} which yields a filtration $X^0 \subset X^1 \subset \cdots \subset X$ such there is a pushout diagram \begin{equation*} \begin{tikzcd} \coprod S^{n-1} \ar{r} \ar[hook]{d} & X^{n-1} \ar[hook]{d} \\ \coprod D^n \ar{r} & X^n \end{tikzcd} \end{equation*} One could take this as an alternative definition of a CW-complex by a filtration with the pushout property. The cells can be recovered as connected components of $X^n\setminus X^{n-1}$. We have \begin{align*} C_i^{CW}(X) = H_i(X^i, X^{i+1}) \xleftarrow{\cong} H_i(\coprod D^i, \coprod S^{i-1}) \cong \bigoplus H_i(D^i, S^{i-1}) \cong \bigoplus \Z^{\# i\text{-cells}}, \end{align*} where the first isomorphism $\leftarrow$ is given by excision. The boundary maps $C_i^{CW}(X) \xrightarrow{\partial} C_{i-1}^{CW}(X)$ come from \begin{align*} H_i(X^i,X^{i-1}) \to H_{i-1}(X^{i-1}) \to H_{i-1}(X^{i-1},X^{i-2}). \end{align*} Under this isomorphism, the matrix entry belonging to $(e,f)$ where $e$ is an $n$-cell, $f$ an $(n-1)$-cell is the \CmMark{degree} of the map. \begin{align*} S^{i-1} \xrightarrow{\phi_e|_{S^{n-1}}} X^{i-1} \xrightarrow{\operatorname{proj}} X^{i-1}/(X^{i-1}\setminus f) \xleftarrow{\phi_f, \cong} D^{i-1}/S^{i-2} \cong S^{i-1}. \end{align*} \begin{expl*} Consider the torus as an identification square. We convince ourselves that the cellular chain complex is given as $\Z \to \Z \oplus \Z \to \Z$, where $1 \mapsto (0, 0)$, since it is described by a map $S^1 \to S^1$ traversing the 2-cell according to orientation has degree $0$. \end{expl*} \subsection{Lefschetz number} Recall that a map $f \colon X \to P$ between CW-complexes is \CmMark{cellular}, if $f(X^i) \subset Y^i$ for all $i$. \begin{thm*}[Cellular approximation] Any map between CW-complexes is homotopic to a cellular map. \end{thm*} \begin{dfn*} The \CmMark{Lefschetz number} of a self-map $f \colon X \to X$ of a finite CW-complex is defnined as \begin{align*} \Lambda(f) = \sum_{i \geq 0} (-1)^i\tr C_i^{CW}(f) \in \Z. \end{align*} \end{dfn*} \begin{rem*} $\Lambda(\id_X) = \chi(X)$. \end{rem*} The following theorem yields a description of Lefschetz numbers by homology. \begin{thm*} $\Lambda(f) = \sum_{i \geq 0}(-1)^i \tr H_i(f)$. \end{thm*} Thus, this number only depends on the homotopy class of $f$. \begin{proof} Similar to the proof of Euler-Poincaré using the additivity of the trace, i.e. in the situation \begin{tikzcd}[row sep=small] 0 \ar{r} & A \ar{r} \ar{d}{a} & B \ar{r} \ar{d}{b} & C \ar{r} \ar{d}{c} & 0\\ 0 \ar{r} & A \ar{r} & B \ar{r} & C \ar{r} & 0 \end{tikzcd} we have $\tr(b) = \tr(a) + \tr(c)$. \end{proof} \begin{thm*} If $f$ has no fixed point, then $\Lambda(f) = 0$. \end{thm*} Jan-Bernhard Kordaß committed Oct 25, 2016 145 Jan-Bernhard Kordaß committed Oct 25, 2016 146 147 148 \begin{rem*} The converse is not true (think of counterexamples, e.g. $S^1 \wedge S^1$), although there is one in the case of simply-connected closed manifolds. \end{rem*} Jan-Bernhard Kordaß committed Oct 25, 2016 149 Jan-Bernhard Kordaß committed Oct 25, 2016 150 151 152 153 154 155 156 157 \begin{proof} Let $X$ be metrizable and let $d$ be a metric. If $X$ is compact, there exists an $\varepsilon > 0$ with $d(f(x), x) > 3\varepsilon$. One can refine'' the CW-structure to a new one such that every cell has diameter $< \varepsilon$. By cellular approximation we can see that there exists a cellular map $g \colon X \to X$ with $g \simeq f$ and $d(g(x),f(x)) < \varepsilon$. Thus $g(\overline e) \cap \overline e = \emptyset$ for every cell $e$. Hence, the diagonal matrix entries of each $C_i^{CW}(g)$ are zero and thus $\Lambda(g) = \Lambda(f) = 0$. \end{proof} Jan-Bernhard Kordaß committed Oct 25, 2016 158 Jan-Bernhard Kordaß committed Oct 25, 2016 159 Jan-Bernhard Kordaß committed Nov 08, 2016 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 \section{Whitehead torsion} \subsection{Introduction/Motivation} Given a homotopy equivalence $f \colon X \xrightarrow{\simeq} Y$ of finite CW-complexes, Whitehead torsion is an assignment $\tau(f) \in \Wh(\pi_1(Y))$ living in the so-called Whitehead group. \begin{thm*}[Properties of Whitehead torsion] \begin{enumerate}[label=(\arabic*)] \item homotopy invariance \item\footnote{This is a deep theorem of Chapman.} If $f \colon X \to Y$ is a homeomorphism, then $\tau(f) = 0$. \item additivity: A cellular pushout is a diagram \begin{equation*} \begin{tikzcd} X_0 \ar{r}{f} \ar[hook]{d}{i} & X_2 \ar{d}\\ X_1 \ar{r} & X \end{tikzcd} \end{equation*} with $X_i$ be CW-complexes, where $f$ is cellular and $i$ is an inclusion of a subcomplex. If the diagram \begin{equation*} \begin{tikzcd} X_0 \ar{rr} \ar[hook]{dd} \ar{rd}{f_0}[swap]{\simeq} &[0.8cm] &[-0.2cm] X_2 \ar[bend left=10]{rd}{f_2}[swap]{\simeq} &[0.8cm] \\ & Y_0 \ar[near start]{rr}{\phi} \ar[near end,hook]{dd}{j} & & Y_2 \ar{dd}{i}\\ X_1 \ar[crossing over]{rr} \ar[bend right=10]{rd}{f_1}[swap]{\simeq} & & X \ar[dashed]{rd}{f} \ar[leftarrow,crossing over]{uu} & \\ & Y_1 \ar{rr}{\psi} & & Y \end{tikzcd} \end{equation*} Roman Sauer committed Nov 21, 2016 188 is a map of cellular pushouts such that $f_i$ are homotopy equivalences. Jan-Bernhard Kordaß committed Nov 08, 2016 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 Then $f$ is a homotopy equivalence and \begin{align*} \tau(f) & = "\tau(f_1) + \tau(f_2) - \tau(f_0)"\\ & = \psi_{*}(\tau(f_1)) + j_{*}(\tau(f_2)) - (\psi \circ i)_{*}(\tau(f_0)) \in \Wh(\pi_1(Y)). \end{align*} A similar additivity holds for the Lefschetz number.\footnote{Idea. $0 \to C_1(X_0) \to C_{*}(X_1) \oplus C_{*}(X_2) \to C_{*}(X) \to 0$ exact.} \item composition formula. If we have $\begin{tikzcd} X \ar{r}{f}[swap]{\simeq} & Y \ar{r}{g}[swap]{\simeq} & Z \end{tikzcd}$, then \begin{align*} \tau(g \circ f) & = "\tau(f) + \tau(g)"\\ & = g_{*}(\tau(f) + \tau(g)) \in \Wh(\pi_1Z). \end{align*} \end{enumerate} \end{thm*} \begin{thm*}[s-cobordism theorem (Mazur, Barden, Stallings, Smale)] Let $M$ be a closed smooth manifold of dimension $\geq 5$. Let $(W, i, j)$ be an s-cobordism \begin{figure}[h!] \centering \begin{tikzpicture}[scale=0.8] \coordinate (L1) at (2,0); \coordinate (L2) at (2,3); \coordinate (LL1) at ($(L1)+(-4,0)$); \coordinate (LL2) at ($(L2)+(-4,0)$); \coordinate (R1) at (8,0.5); \coordinate (R2) at (8,2); \coordinate (RR1) at ($(R1)+(3,0)$); \coordinate (RR2) at ($(R2)+(3,0)$); % left inclusion \ellipsebetweenvert{LL1}{LL2} \node[below] at (LL1) {$M$}; \draw[right hook->] ($(LL1) + (1.2,1.5)$) to node[above] {$i$} node[below]{$\simeq$} ($(L1) + (-1.2,1.5)$); % cobordism \ellipsebetweenvert{L1}{L2} \node[below] at (L1) {$M_0$}; \draw[out=0,in=180] (L1) to (5,-1) to (R1); \draw[out=0,in=180] (L2) to (R2); \topgenus[0.35]{5,0} \topgenus[0.22]{6,1.2} \node at (6,3) {$W$}; \ellipsebetweenvert[left]{R1}{R2} \node[below] at (R1) {$M_1$}; % right inclusion \draw[left hook->] ($(RR1) + (-1,0.75)$) to node[above] {$j$} node[below]{$\simeq$} ($(R1) + (1,0.75)$); \ellipsebetweenvert{RR1}{RR2} \node[below] at (RR1) {$N$}; \end{tikzpicture} % sketch: \includegraphics[width=0.8\textwidth]{img/1.png} \end{figure} i.e. $\partial W = M_0 \coprod M_1$ and $i \colon M \hookrightarrow W$, $j \colon N \hookrightarrow W$ are homotopy equivalences. Then $\tau(M \xrightarrow{i} W) = 0$ if and only if $(W,i_0,i_1)$ is trivial, i.e. \begin{figure}[h!] \centering \begin{tikzpicture}[every node/.style={scale=0.8}] \coordinate (LU1) at (2.4,1.5); \coordinate (LU2) at (2.4,2.5); \coordinate (RU1) at (5,1.3); \coordinate (RU2) at (5,2.3); \coordinate (LD1) at (2.4,0.5); \coordinate (LD2) at (2.4,-0.5); \coordinate (RD1) at (5,0.5); \coordinate (RD2) at (5,-0.5); % leftmost part \node at (0,3) {$\exists$}; \node[scale=1.5] at (-1,1) {$M$}; \draw[right hook->] (-0.4,1.2) to node[above] {$i$} (2,2); \draw[right hook->] (-0.4,0.8) to node[below] {$\operatorname{incl}$} (2,0); % upper cobordism \ellipsebetweenvert{LU1}{LU2} \draw[out=0,in=180] (LU1) to (3,1.6) to (RU1); \draw[out=0,in=180] (LU2) to (4,2.2) to (RU2); \ellipsebetweenvert[left]{RU1}{RU2} % arrow from the cobordism to the cylinder \draw[->] ($(LU1)!0.5!(RU1) + (0,-0.1)$) to node[left]{$\cong$} node[right]{diffeo} ($(LU1)!0.5!(RU1) + (0,-0.7)$); % cylinder \ellipsebetweenvert{LD1}{LD2} \draw (LD1) to (RD1); \draw (LD2) to (RD2); \ellipsebetweenvert[left]{RD1}{RD2} \end{tikzpicture} % sketch: \includegraphics[width=0.5\textwidth]{img/2.png} \end{figure} \end{thm*} This theorem implies the Poincaré conjecture in dimensions at least 6, which says that if $M$ is a closed (smooth) manifold that is homotopy equivalent to $S^n$, then $M$ is homeomorphic to $S^n$. The proof of this implication is basically along these lines: Pick disjoint embedded $n$-disks in $M$ and remove them. The result is a manifold $W$ with two boundary components. Consider the s-cobordism theorem for this manifold as depicted in the figure below, where $f$ is a diffeomorphism of $(n-1)$-spheres. \begin{figure}[h!] \centering \begin{tikzpicture} \coordinate (U1) at (0,3); \coordinate (U2) at (2.2,3); \coordinate (D1) at (-0.2,0); \coordinate (D2) at (1.8,0); % bordism on the left \ellipsebetweenhor{U1}{U2} \draw[out=270,in=90] (U1) to (0.3,1.8) to (D1); \draw[out=270,in=90] (U2) to (2.3,1.7) to (D2); \ellipsebetweenhor[upper]{D1}{D2} \node[left] at (0,1.5) {$W$}; % arrows and lower disk \draw[bend left=15,->] (2.8,3.3) to node[above] {$f$} (4.2,3.3); \draw[->] (3,1.5) to node[above] {$\cong$} (4,1.5); \draw[left hook->] (2.5,-1) to (1.3,-0.5); \draw[right hook->] (4.2,-1) to (5.5,-0.5); \draw[pattern=north west lines,pattern color=gray] (3.35,-1) ellipse (0.6 and 0.3); \node at (3.8,-0.4) {$D^n$}; % cylinder on the right \ellipsebetweenhor{5,3}{7,3} \draw (5,3) to (5,0); \draw (7,3) to (7,0); \ellipsebetweenhor[upper]{5,0}{7,0} \node[right] at (7,1.5) {$S^{n-1} \times [0,1]$}; \end{tikzpicture} % sketch: \includegraphics[width=0.7\textwidth]{img/3.png} \end{figure} By filling top and bottom, the Poincaré conjecture is implied, if we can extend a diffeomorphism $f \colon S^{n-1} \xrightarrow{\cong} S^{n-1}$ to a homeomorphism $F \colon D^n \xrightarrow{\cong} D^n$. This can be done by the so-called Alexander trick ($F(t x) = tf(x)$ for $t \in [0,1], x\in S^{n-1}$). \subsection{Whitehead group and lower K-theory} Let $R$ be a unital ring. Then \begin{align*} K_0(R) := \left< G \mid R \right>_{\text{ab}} \end{align*} with generators $G = \{$ isomorphism classes $[P]$ of fin. gen. projective $R$-modules $\}$ and relations $R = \{\ [P_1] = [P_0] + [P_2]$ whenever $0 \to P_0 \to P_1 \to P_2 \to 0$ is exact $\}$. (Recall that a direct summand of in a free $R$-module is called a \CmMark{projective module}.) This can be understood as a kind of universal dimension for projective $R$-modules. \begin{align*} K_1(R) := \left< G \mid R \right>_{\text{ab}} \end{align*} with generators $G = \{$ conjugacy classes $[f]$ of automorphisms $f \colon P \to P$ of fin. gen. projective $R$-modules $\}$ and relations $R$ given as follows. \begin{enumerate}[label=(\roman*)] \item Every commuting diagram \begin{equation*} \begin{tikzcd} 0 \ar{r} & P_0 \ar{r} \ar{d}{f_0}[swap]{\cong} & P_1 \ar{r} \ar{d}{f_1}[swap]{\cong} & P_2 \ar{r} \ar{d}{f_2}[swap]{\cong} & 0\\ 0 \ar{r} & P_0 \ar{r} & P_1 \ar{r} & P_2 \ar{r} & 0. \end{tikzcd} \end{equation*} gives rise to a relation $[f_1] = [f_0] + [f_2]$. \item $f,g \colon P \xrightarrow{\cong} P$ yield a relation $[f \circ g] = [f] + [g]$. \end{enumerate} This can be understood as an attempt to define a universal determinant of an automorphism. There is a more common definition of $K_1(R)$ in terms of the general linear groups with coefficients in $R$. Recall that $\GL(R) = \colim_{n \to \infty} \GL_n(R)$ with respect to the inclusion $\GL_n(R) \hookrightarrow \GL_{n+1}(R)$ to the upper left block. Then \begin{align*} K_1(R) = \GL(R)_{\text{ab}} = \GL(R)/[\GL(R),\GL(R)]. \end{align*} The so-called \CmMark{Whitehead lemma} states that $[\GL(R),\GL(R)] = E(R)$, where $E(R)$ is the subgroup of $\GL(R)$ generated by all elementary upper triangular matrices with ones on the diagonal. As a consequence, if $R$ is a field then the determinant defines an isomorphism $\det \colon K_1(R) \xrightarrow{\cong} \R\setminus\{0\}$. To see the equivalence of these two definitions, we can use the map \begin{align*} \GL(R) & \to K_1(R)\\ A & \mapsto [R^n \to R^n, \ x \mapsto Ax] \end{align*} and the fact that is descents to $\GL(R)_{\text{ab}} \to K_1(R)$. The inverse homomorphism is given by $R^n \cong \{ P \oplus Q \xrightarrow{f \otimes \id} P \oplus Q \mid f \text{ iso } \}$. \begin{dfn*} Let $\Gamma$ be a group. Then the \CmMark{Whitehead group} is defined as \begin{align*} \Wh(\Gamma) := \coker(\Gamma \times \{\pm 1\} \to K_1(\Z[\Gamma]), \ (\gamma, \pm 1) \mapsto \pm[\gamma]) \end{align*} \end{dfn*} \begin{expl*} The Whitehead group $\Wh(\{1\})$ is trivial, since the determinant yields an isomorphism $K_1(\Z) \xrightarrow{\det} \{\pm 1\}$. It is a conjecture that torsion-free groups $\Gamma$ have vanishing Whitehead group. Assertion: $\Wh(\Z/5) \cong \Z$. Here only prove that $\Wh(\Z/5)$ is infinite. We have a map $\phi_{*} \colon K_1(\Z[\Z/5]) \to K_1(\C)$ induced by $\Z[\Z/5] \xrightarrow{\phi} \C, \ t \mapsto \xi$, where $\Z/5 \cong \left$ and $\xi = \exp(2\pi i/5) \in \C$. Thus \begin{equation*} \begin{tikzcd} K_1(\Z[\Z/5]) \ar{r}{\phi_{*}} \ar{d} & K_1(\C) \ar{r}{\det} & \C^{\times} \ar{r}{|\blank|} & \R_{> 0}\\ \Wh(\Z/5) \ar[bend right=10]{rrru}{\tau} & & & \end{tikzcd} \end{equation*} One can see that $1 - t - t^{-1}$ is a unit in $\Z[\Z/5]$, since $(1 - t - t^{-1})( - t^2 - t^3) = 1$ and thus $\tau([1 - t - t^{-1}]) \neq 1$ \end{expl*} Jan-Bernhard Kordaß committed Oct 25, 2016 445 446 447 \chapter{Harmonic Maps [Andy Sanders]} Jan-Bernhard Kordaß committed Nov 08, 2016 448 Also consider the notes \url{www.mathi.uni-heidelberg.de/~asanders/harmonicmaps.htm}. Jan-Bernhard Kordaß committed Nov 08, 2016 449 Jan-Bernhard Kordaß committed Oct 25, 2016 450 451 \section{Basics of harmonic maps} Jan-Bernhard Kordaß committed Nov 08, 2016 452 453 In the following let every manifold be oriented (for integration safety reasons). Jan-Bernhard Kordaß committed Oct 25, 2016 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 \subsection{Background differential geometry} Let $E \to M$ be an $\R$-vector bundle over $M$ (second countable, hausdorff manifold) of rank $r$. A \CmMark{connection} $\nabla$ on $E$ is an $\R$-linear map \begin{align*} \nabla \colon \Omega^0(E) \to \Omega^0(\T^{*}M \otimes_{\R} E) =: \Omega^1(M,E), s \mapsto \nabla_{\blank} s \end{align*} where $\Omega^0(E)$ denotes smooth sections in $E$, such that \begin{enumerate} \item $\nabla_{X+Y}s = \nabla_Xs + \nabla_Ys$, \item $\nabla_X(s+s') = \nabla_X s + \nabla_Xs'$ \item $\nabla_{fX} s = f\nabla_Xs$ \item $\nabla_X(fs) = f\nabla_Xs + X(f) s$. \end{enumerate} Let $q$ be an inner product on $E$. We say that $\nabla$ is a \CmMark{metric connection} for $q$, if for all $s,t \in \Omega^0(E)$ we have \begin{align*} Jan-Bernhard Kordaß committed Nov 08, 2016 473 \dop q(s,t) = q(\nabla s,t) + q(s, \nabla t). Jan-Bernhard Kordaß committed Oct 25, 2016 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 \end{align*} \begin{expl*} Let $(M,g)$ be a riemannian manifold with tangent bundle $E = \T M$ and Levi-Civita connection $\nabla$ of $g$. Let $X,Y \in \Omega^0(M)$ be vector fields, i.e. $X = X^i \frac{\partial}{\partial x^i}$ and $Y = Y^j \frac{\partial}{\partial x^j}$ in local co-ordinates. (Abbreviate $\partial_i$ for $\frac{\partial}{\partial x^i}$.) \begin{align*} \nabla_XY = \nabla_{X^i\partial_i} Y^i\partial_i = X^i(\nabla_{\partial_i}Y^i\partial_i) = X^i(\partial_iY^i\partial_i + Y^i\nabla_{\partial_i}\partial_i) = X^i(\partial_iY^i\partial_i + Y^i\Gamma_{ij}^k\partial_i) \end{align*} where $\Gamma_{ij}^k = g^{km}(\partial_ig_{im} + \partial_j g_{im} - \partial_mg_{ij})$ for $g_{ij} = g(\partial_i,\partial_j)$ and $g^{km}$ is the $km$-entry of $g^{-1}$. \end{expl*} Out of $E$ one can build another bundle $E^{*} = \Hom(E,\R)$ and given another vector bundle $F$, one can build $\Hom(E,F)$, \begin{dfn*} Let $(E,\nabla) \to M$ be a vector bundle with a connection over $M$. The space of \CmMark{$p$-forms} on $m$ with values in $E$ is the $C^{\infty}(M)$-module $\Omega^p(M,E) = \Omega^0(M,\bigwedge^p\T^{*}M \otimes E)$. Elements $\alpha$ in $\Omega^p(M,E)$ have representations as linear combination of $\alpha_{i_1,\cdots,i_p}\dop x^{i_1} \wedge \cdots \wedge \dop x^{i_p} \otimes (s_1, \cdots s_p)$. \end{dfn*} \begin{dfn*} The exterior covariant derivative is the map given by extension of \begin{align*} \dop^{\nabla} \colon \Omega^p(M,E) & \to \Omega^{p+1}(M,E),\\ \alpha \otimes u & \mapsto \dop^{\nabla}(\alpha \otimes u) = \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u \end{align*} for $\alpha \in \bigwedge^p\T^{*}M$, $u \in \Omega^0(E)$. \end{dfn*} We want to define an inner product on $\Omega^p(M,E)$. For this, fix a metric $g$ on $M$ and let $(E,\nabla,q) \to M$ be a vector bundle with metric and connection over $M$. \begin{align*} \left< \alpha \otimes u, p \otimes v\right> = \int_M g(\alpha,p) q(u,v) \dop v_g \end{align*} is a number. (For this integral to be finite, assume $M$ is compact or work with compactly supported sections.) \begin{dfn*} The \CmMark{exterior covariant codifferential}\footnote{non-standard notation} is the formal $L^2$-adjoint of $d$ \begin{align*} \delta^{\nabla} \colon \Omega^p(M,E) \to \Omega^p(M,E) \end{align*} such that $\left< \dop^{\nabla}(\alpha \otimes u), \beta \otimes v\right> = \left<\alpha \otimes u, \delta^{\nabla}(\beta \otimes v)\right>$. \end{dfn*} \begin{rem*}[Fact] An integration by parts arguement shows that $\delta^{\nabla}$ exists and, when $\nabla$ is a metric connection, then \begin{align*} \delta^{\nabla} \colon \Omega^1(M,E) \to \Omega^0(M,E), \ \alpha \otimes u \mapsto -\tr_g(\nabla^{\T^{*} \otimes E} \alpha \otimes u), \end{align*} where for $\Omega^1(M,E) \to \Omega^0(M, \T^{*}M \otimes \T^{*}M \otimes E)$, we can take a trace with the metric by choosing an orthonormal basis. \end{rem*} \begin{dfn*} A \CmMark{harmonic $p$-form} with values in $E$ is an element $\omega_i \in \Omega^p(M,E)$ such that $\delta^{\nabla} = \delta^{\nabla} \omega = 0$. As a matter of fact this is equivalent to $\Delta \omega = 0$ for $\Delta := \delta^{\nabla} \circ \dop^{\nabla} + \dop^{\nabla} \circ \delta^{\nabla}$ (Consider $\left<\Delta \omega, \omega\right>$ and utilize the obvious stuff). \end{dfn*} \subsection{Definition of harmonic maps of 1st variation formula} Let $(M,g)$ and $(N,h)$ be two riemannian manifolds and let $f \colon M \to N$ be a smooth map. Then $\dop f \colon \T M \to \T N$ is an element $\dop f \in \Omega^0(\Hom(\T M, f^{*}\T N)) = \Omega^0(\T^{*}M \otimes f^{*}\T N)$. Next, the metrics $g,h$ induce a metric on $\T^{*}M \otimes f^{*}\T N$. \begin{dfn*} The energy density of $f \colon M \to N$ is $e(f) := \frac{1}{2} \left< \dop f, \dop f\right>_{\T^{*}M \otimes f^{*}\T N} = \frac{1}{2} \|\dop f\|^2$. \end{dfn*} Choose co-ordinates $\{x^i\}$ in $M$ and $\{y^i\}$ in $N$. With respect to these, we have \begin{align*} \frac{1}{2} \|\dop f \|^2 = \frac{1}{2}y^{ij} \partial_if^{*}\partial_jf^{\beta}h_{\alpha\beta}(f). \end{align*} \begin{dfn*} The \CmMark{Dirlichlet energy} is given by \begin{align*} E \colon C^2_0(M,N) \to \R, \ f \mapsto \int_M e(f) \dop V_g. \end{align*} A \CmMark{critical map} (or \CmMark{stationary map}) is a map $f \colon M \to N$ such that for all compactly supported $F \colon M \times (-\varepsilon, \varepsilon) \to N$ $C^2$-map (variation of $f$) with $F(x,0) = f(x)$ we have that \begin{align}\label{eq:first-variation} \delta E(\nu) := \left.\frac{\dop}{\dop t} E(F) \right|_{t = 0} = 0 \end{align} for $\nu = \frac{\dop}{\dop t} F|_{t = 0} \in \Omega^0(f^{*}\T N)$. The \cref{eq:first-variation} is called \CmMark{first variation in the direction of $\nu$}. \end{dfn*} \begin{dfn*} The map $f \colon (M,g) \to (N,h)$ is called \CmMark{harmonic}, if it is a critical point for the Dirlichlet energy. \end{dfn*} \begin{dfn*} Let $\dop f \in \Omega^1(M, f^{*}\T N)$ then $\nabla \dop f \in \Omega^0(M, \T^{*}M \otimes \T^{*}M \otimes E)$. The \CmMark{second fundamental form} of $f$ is $\nabla \dop f := B_f$, which is a symmetric $2$-tensor on $M$. \end{dfn*} \begin{dfn*} The \CmMark{tension field} of $f$ is the trace of $B_f$: $\tau(f) := \tr_g(B_f) \in \Omega^0(M,f^{*}\T N)$. \end{dfn*} \begin{thm*}[1st variation of $E$] Let $F \colon M \times (\varepsilon, \varepsilon) \to N$ a variation of $f$ and let $\nu = \frac{\dop}{\dop t}F|_{t = 0}$. Then \begin{align*} \delta E(\nu) = \frac{\dop}{\dop t}E(F)|_{t = 0} = - \int_M \left<\tau(f), \nu\right> \dop v_g. \end{align*} \end{thm*} \begin{proof} The variation $F \colon M \times (-\varepsilon,\varepsilon) \to N$ yields a pullback connection on $F^{*}\T N$, which shows \begin{align*} \frac{\dop}{\dop t}E(F)|_{t = 0} & = \frac{1}{2} \int_M \frac{\dop}{\dop t}\left<\dop F, \dop F\right> \dop V_g|_{t = 0} = \int_M \left<\nabla_{\frac{\partial}{\partial t}}\dop F, \dop F\right> \dop V_g|_{t = 0}\\ & = \int_M \left<\nabla^{f^{*}\T N}\nu, \dop f\right> \dop V_g \overset{(*)}{=} \int_M \left<\nu, \delta^{\nabla^{f^{*}\T N}} \dop f\right> \dop V_g\\ & = -\int_M \left< \nu, \tr_g(\nabla \dop f) \right> \dop V_g Jan-Bernhard Kordaß committed Nov 08, 2016 602 603 604 = - \int_M \left< \nu, \tau(f)\right> \dop V_g, \end{align*} where $(*)$ follows by a calculation in local co-ordinates. Jan-Bernhard Kordaß committed Oct 25, 2016 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 \end{proof} \begin{cor*}[Fundamental theorem of the calculus of variations] A $C^2$-map $f \colon (M,g) \to (N,h)$ is harmonic if and only if $\tau(f) = 0$. \end{cor*} What does $\tau(f) = 0$ look like? Fix local co-ordinates $\{x^i\}$ on $M$ and $\{y^j\}$ on $N$. Then $\dop f = \partial_if^{alpha} \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}$ and thus \begin{align*} \nabla \dop f & = \nabla \partial_i f^{\alpha} \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} = \partial_j\partial_i f^{\alpha} \dop x^j \otimes \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} + \partial_if^{\alpha} \nabla \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}}\\ & = A + \partial_i f^{\alpha}(\nabla \dop x^i \otimes \frac{\partial}{\partial y^{\alpha}} + \dop x^i \otimes \nabla \frac{\partial}{\partial y^{\alpha}})\\ & = A + \partial_i f^{\alpha}( -\Gamma^i_{jk} \dop x^i \otimes \dop x^k \otimes \frac{\partial}{\partial y^{\alpha}} + \dop x^i \partial_j f^{\beta} \Gamma^{\gamma}_{\alpha\beta} \frac{\partial}{\partial y^{\gamma}})\\ & = \partial_i \partial_jf^{\gamma} \Gamma_{ij}^k \partial_k f^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_jf^{\alpha}\partial_if^{\beta)} \dop x^i \otimes \dop x^j \otimes \frac{\partial}{\partial y^j}. \end{align*} Thus $\tau(f) = (\Delta_gf^{\gamma} + \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})$. Jan-Bernhard Kordaß committed Nov 08, 2016 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 \section{Example and the Bochner formula (a glimpse of rigidity)} Recall that above we considered $C^2$-maps $f \colon (M,g) \to (N,h)$ with tension field \begin{align*} \tau(f) := \tr_g(\nabla \dop f) = 0 \in \Omega^0(M,f^*\T N). \end{align*} In local co-ordinates $\{x^i\}$ on $M$ and $\{y^{\alpha}\}$ on $N$ this means \footnote{Use roman indices for the $M$ and greek ones for $N$.} \begin{align*} \tau(f)^{\gamma} \frac{\partial}{\partial y^{\gamma}} = (\Delta_g f^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(f) \partial_if^{\alpha}\partial_jf^{\beta}g^{ij})\partial_{\gamma} = 0, \end{align*} where $\tilde \Gamma$ are the Christoffel symbols on $(N,h)$. \begin{expl*} \begin{enumerate}[label=\Roman*.] \item Let $(M,g) = (\R, \dop t^2)$ and let $\eta \colon \R \to (N,h)$. From above we know that the Laplace-Beltrami operator here reads \begin{align*} \Delta_gf = g^{ij} (\partial_i\partial_jf - \Gamma_{ij}^k\partial_kf) = g^{ij}(\partial_i\partial_jf) = \partial_t^2f \end{align*} for \begin{align*} \Gamma_{ij}^k = \frac{g^{km}}{2}(\partial_ig_{im} + \partial_jg_{im} - \partial_m(g_{ij}) \end{align*} ($=0$ in $\{g_{ij}\}$ is constant)\todo{repair} and $\{g_{ij}\} = g_{11} = f(\partial_t,\partial_t) = \dop t^2(\partial_t,\partial_t) = 1$. Hence \begin{align*} \tau(\eta)^{\gamma}\partial_{\gamma} = (\ddot \eta^{\gamma} + \tilde \Gamma_{\alpha\beta}^{\gamma}(\eta) \dot\eta^{\alpha}\dot\eta^{\beta})\partial_{\gamma} = 0, \end{align*} which is if and only if $\eta$ is a geodesic, i.e. the covariant derivate along $M$ of the curves speed vanishes: $\frac{\Dop}{\dop t} \dot \eta = 0$ and thus $E(\eta)|_a^b = \frac{1}{2}\int_a^b\|\dot\eta\|^2\dop t$. \item Now let $f \colon (M,g) \to \R$. Here $\tau(f) = \Delta_gf = 0$. \begin{prop*} If $M$ is closed, then the energy harmonic functions are constant. \end{prop*} \begin{proof} By Green's theorem (integration by parts) \begin{align*} \int_M \underbrace{g(\nabla f, \nabla f)}_{=\|\nabla f\|^2} \dop V_g = - \int \Delta_g f \cdot f \dop V_g = 0 \end{align*} for $\dop V_g = \sqrt{\det (\{g_{ij}\})} \dop x^1 \wedge \cdots \wedge \dop x^n$. Thus $\|\nabla f\|^2 = 0$ and $f$ must be constant. \end{proof} In our example, this shows $\Delta_g f = \lambda f$. \item Let $f \colon (M,g) \to (N,h)$ be an isometric immersion, i.e. $\dop f$ is injective and $g = f^{*}h = h(\dop f, \dop f)$. Then we have \begin{align*} e(f) & = \frac{1}{2} \|\dop f\|^2 = \frac{1}{2}h_{\alpha\beta} \partial_if^{\alpha}\partial_jf^{\beta}g^{ij} = \frac{1}{2}\partial_if^{\alpha}\partial_jf^{\beta}h(\partial_{\alpha},\partial_{\beta}) g^{ij}\\ & = \frac{1}{2}h(\partial_if^{\alpha}\partial_{\alpha},\partial_jf^{\beta}\partial_{\beta})g^{ij} = \frac{1}{2}h(\dop f(\partial_i),\dop f(\partial_j)) g^{ij} = \frac{1}{2}g_{ij}g^{ij} = \frac{m}{2} \end{align*} and hence $E(f) = \frac{m}{2}\Vol(f)$, where $\Vol(f) = \int_M\dop V_{f^{*}h} = \int_M\dop V_g$. This shows that $f$ is critical for $E$ if and only if $f$ is critical for $\Vol \colon \Imm(M,N) \to \R_+$. The latter is clearly if and only if $f$ is a \textbf{minimal submanifold}. Examples of minimal submanifolds in $\R^3$ include the 2-plane, or the helicoid. \end{enumerate} \end{expl*} \subsection{Composition laws for harmonic maps} Consider the composition \begin{align*} (M,g) \xrightarrow{f} (N,h) \xrightarrow{u} (Z,b). \end{align*} In general, if $f,u$ are harmonic, this needs not be harmonic again, which can be considered a bug or a feature''. \begin{align*} B_{u \circ f}(X,Y) = B_u(\dop f(X), \dop f(Y)) + \dop u (B_f(X,Y)) \end{align*} for $X,Y \in \T_pM$ and thus $B_{u \circ f} = \nabla^{\T^{*}M \otimes (u \circ f)^{*}\T N}(\dop(u \circ f))$. Hence $\tau(u \circ f) = \dop (\tau(f)) + \tr_g(f^{*}B_u)$. If $f$ is harmonic, then $\tau(u \circ f) = \tr_g(f^{*}B_u)$. \begin{prop*} If $f \colon M \to N$, is harmonic and $u \colon N \to Z$ is totally geodesic, i.e. $B_u = 0$. Then $u \circ f$ is harmonic. \end{prop*} What if $u \colon N \to \R$ is a function and $f$ is harmonic? Then \begin{align*} \tau(u \circ f) = \tr_g(f^{*}B_u) = \tr_g(f^{*}(\Hess(u)) = \sum_{i = 1}^n f^{*}(\Hess(u)) (E_i,E_i). \end{align*} Recall that a function $u \colon (N,h) \to \R$ is convex, if $\Hess(u)$ is positive definite. If $f$ is harmonic and $u$ is convex, then $\tau(u \circ f) = \nabla_g u \circ f \geq 0$ (these are called \CmMark{subharmonic functions}). \begin{thm*} A map is harmonic if and only if it pulls back germs of convex functions to germs of subharmonic functions. \end{thm*} There are various useful applications of the synthetic view'' on harmonic functions (e.g. Gromov-Shane). \begin{thm*} Suppose $(M,g)$ is closed, connected and $(N,h)$ is $1$-connected with non-positive curvature. Then every harmonic map $f \colon (M,g) \to (N,h)$ is constant. \end{thm*} \begin{proof} The distance function $N \to \R_{\geq 0}, x \mapsto \dop_N(p,x)^2$ for every $p \in N$ is actually smooth and strictly convex, e.g. $\dop_{\R^n}(0,x)^2 = x_1^2 + \cdots + x_n^2$. In case $f$ is harmonic, we have \begin{align*} \Delta_gu \circ f = \tau(u \circ f) = \tr_g(f^{*}B_u) \geq 0 \end{align*} and \begin{align*} -\int \| \dop(u \circ f)\|^2 \dop V_g = \int_M \Delta_gu \circ f \dop V_g \geq 0. \end{align*} Thus $\|\dop (u \circ f)\| = 0$ and hence $u \circ f$ is constant. \end{proof} \subsection{Bochner formulas} Let $(E,\nabla,a)$ be a riemannina vector bundle, i.e. $a$ is a metric on $E$, $\nabla$ is a connection on $E$ preserving $a$ ($\nabla a = 0$) and there is a vector bundle projection map $E \to (M,g)$. Let $\omega \in \Omega^p(M,E)$ and let $\nabla$ be a connection on $\Omega^p(M,E)$. \begin{align*} \hat\nabla \colon \Omega^p(M,E) \to \Omega^p(M,\T^{*}M \otimes \T^{*}M \otimes E), \quad \omega \mapsto ( (X,Y) \mapsto \nabla_X\nabla_Y\omega - \nabla_{\nabla_XY}\omega ) \end{align*} The \CmMark{trace Laplacian} is the operator \begin{align*} \nabla^2 \colon \Omega^p(M,E) \to \Omega^p(M,E), \quad \omega \mapsto \tr_g(\hat\nabla \omega). \end{align*} Recall that the \CmMark{Hodge Laplacian} was the operator \begin{align*} \dop^{\nabla} \colon \Omega^p(M,E) \to \Omega^{p+1}(M,E), \quad \alpha \otimes u \mapsto \dop \alpha \otimes u + (-1)^p \alpha \wedge \nabla u. \end{align*} With respect to the $L^2$-pairing $\beta \otimes v \mapsto \int_Mg(\alpha,\beta) a(u,v)\dop V_g$ it has a formal adjoint \begin{align*} \delta^{\nabla}\colon \Omega^{p+1}(M,E) \to \Omega^p(M,E). \end{align*} The Hodge Laplacian is the degree preserving operator given by $\dop^{\nabla} \circ \delta^{\nabla} + \delta^{\nabla} \circ \dop^{\nabla} =: \Delta_a$. The \CmMark[Bochner-Lichnerowicz formula]{(generalized) Bochner-Lichnerowicz formula} is given by \begin{align*} \nabla_a \omega = - \nabla^2\omega + S_{\omega}. \end{align*} for $S_{\omega} \in \Omega^p(M,E)$ with \begin{align*} S_{\omega}(X_1, \cdots, X_p) = \sum_{k = 1}^p\sum_{i = 1}^m(-1)^k(R^{\tilde \nabla}(e_i, X_k)\omega) (e_i,X_1, \ldots, \hat X_k, \ldots, X_n) \end{align*} for $X_i \in \T_pM$, $m = \dim M$ and $\{e_i\}$ an orthonormal frame around $p$.\footnote{Hat ($\hat X_k$), as always, means to omit the k-th term.} \begin{cor*} Let $f \colon (M,g) \to (N,h)$ be harmonic. Then \begin{align*} \Delta_ge(f) = \|B_f\|^2 - \sum_{ij}\underbrace{h(R^h(f_{*}e_i,f_{*}e_j) f_{*}e_j, f_{*}e_i))}_{= \lambda \sec(e_i,e_j)} + \sum_ih(f_{*}(\Ric^g(e_i)),f_ke_i) \end{align*} for an orthonormal frame $\{e_i\}$. \end{cor*} The key observation for an application of this is that, if $\Ric^g$ is a positive operator, then the latter sum is positive. \begin{thm*}[Eells-Sampson] Let $(M,g)$ be a closed with non-negative Ricci curvature and let $(N,h)$ have non-positive sectional curvature. \begin{enumerate}[label=(\roman*)] \item Then any harmonic map $f \colon (M,g) \to (N,h)$ is totally geodesic, i.e. $\nabla \dop f = B_f = 0$. \item If $\Ric^g$ is positive at any point, then $f$ is constant. \item If the sectional curvature of $(N,h)$ is strictly negative, then $f$ is constant or $f(M)$ is closed geodesic. \end{enumerate} \end{thm*} \begin{proof} The first statement easily follows from the corollary and $\int_M \left< \nabla u, \nabla v\right> \dop V_g = - \int_M \Delta u \cdot v \dop V_g$. The second is also not that hard and the last requires some work. \end{proof} Jan-Bernhard Kordaß committed Oct 25, 2016 808 809 810 811 %%% Local Variables: %%% mode: latex %%% TeX-master: "skript-rtg-lectures-ws1617" %%% End: