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For \( p \in \left( {0,1}\right), G \) a finite graph and \( \beta = - \frac{1}{2}\ln \left( {1 - p}\right) \), we obtain\n\n\[{\mu }_{\beta, G}^{f}\left\lbrack {{\sigma }_{x}{\sigma }_{y}}\right\rbrack = {\phi }_{p,2, G}^{0}\left( {x \leftrightarrow y}\right) ,\]\n\n\[{\mu }_{\beta, G}^{ + }\left\lbrack {\sigma }_{x}\... | Proof We leave the proof as an exercise. | No |
Proposition 3.10. There exists a unique infinite-volume FK-Ising measure with parameter \( {p}_{c} = \sqrt{2}/\left( {1 + \sqrt{2}}\right) \) and there is almost surely no infinite cluster under this measure. Correspondingly, there exists a unique infinite-volume spin Ising measure at \( {\beta }_{c} \) . | Proof As described above, it is sufficient to prove that \( {\phi }_{{p}_{sd},2}^{0} = {\phi }_{{p}_{sd},2}^{1} \) . First note that there is no infinite cluster for \( {\phi }_{{p}_{sd},2}^{0} \) thanks to Exercise 3.6. Via the Edwards-Sokal coupling, the infinite-volume Ising measure with free boundary conditions, de... | No |
Proposition 3.17. Let \( p \in \left( {0,1}\right) \) and let \( \left( {{\Omega }_{\delta }^{\diamond },{a}_{\delta },{b}_{\delta }}\right) \) be a FK Dobrushin domain, then for any configuration \( \omega \) ,\n\n\[ \n{\phi }_{{\Omega }_{\delta }, p}^{{a}_{\delta },{b}_{\delta }}\left( \omega \right) = \frac{1}{Z}{x}... | Proof Recall that\n\[ \n{\phi }_{{\Omega }_{\delta }, p}^{{a}_{\delta },{b}_{\delta }}\left( \omega \right) = \frac{1}{Z}{\left\lbrack p/\left( 1 - p\right) \right\rbrack }^{o\left( \omega \right) }{2}^{k\left( \omega \right) }.\n\]\n\nUsing arguments similar to Proposition 3.3, the dual of \( {\phi }_{{\Omega }_{\delt... | Yes |
Proposition 4.2. There exists \( C > 0 \) such that, for any preharmonic function \( h : {\Omega }_{\delta } \rightarrow \mathbb{C} \) and any two neighboring sites \( x, y \in {\Omega }_{\delta } \) ,\n\n\[ \n\left| {h\left( x\right) - h\left( y\right) }\right| \leq {C\delta }\frac{\mathop{\sup }\limits_{{z \in {\Omeg... | Proof Let \( x, y \in {\Omega }_{\delta } \) . The preharmonicity of \( h \) translates to the fact that \( h\left( {X}_{n}\right) \) is a martingale (where \( {X}_{n} \) is a simple random walk killed at the first time it exits \( {\Omega }_{\delta } \) ). Therefore, for \( x, y \) two neighboring sites of \( {\Omega ... | Yes |
Proposition 4.3. A family \( {\left( {h}_{\delta }\right) }_{\delta > 0} \) of preharmonic functions on the graphs \( {\Omega }_{\delta } \) is pre-compact for the uniform topology on compact subsets of \( \Omega \) if one of the following properties holds:\n\n(1) \( {\left( {h}_{\delta }\right) }_{\delta > 0} \) is un... | Proof Let us prove that the proposition holds under the first hypothesis and then that the second hypothesis implies the first one.\n\nWe are faced with a family of continuous maps \( {h}_{\delta } : \Omega \rightarrow \mathbb{C} \) and we aim to apply the Arzelà-Ascoli theorem. It is sufficient to prove that the funct... | No |
Proposition 4.6. Any limit of a sequence of preharmonic functions on \( {\Omega }_{\delta } \) converging uniformly on any compact subset of \( \Omega \) is harmonic in \( \Omega \) . | Proof Let \( \left( {h}_{\delta }\right) \) be a sequence of preharmonic functions on \( {\Omega }_{\delta } \) converging to \( h \) . Via Propositions 4.2 and 4.3, \( {\left( \frac{1}{\delta }\left\lbrack {h}_{\delta }\left( \cdot + \delta \right) - {h}_{\delta }\right\rbrack \right) }_{\delta > 0} \) is precompact. ... | Yes |
Theorem 4.7. Let \( \Omega \) be a simply connected domain with two marked points a and \( b \) on the boundary, and \( f \) a bounded continuous function on the boundary of \( \Omega \) . Let \( {f}_{\delta } : \partial {\Omega }_{\delta } \rightarrow \mathbb{C} \) be a sequence of uniformly bounded functions convergi... | Proof Since \( {\left( {f}_{\delta }\right) }_{\delta > 0} \) is uniformly bounded by some constant \( M \), the minimum and maximum principles imply that \( {\left( {h}_{\delta }\right) }_{\delta > 0} \) is bounded by \( M \) . Therefore, the family \( \left( {h}_{\delta }\right) \) is precompact (Proposition 4.3). Le... | No |
Proposition 4.9 (Riesz representation formula). Let \( f : {\Omega }_{\delta } \rightarrow \mathbb{C} \) be a function vanishing on \( \partial {\Omega }_{\delta } \) . We have\n\n\[ f = \mathop{\sum }\limits_{{y \in {\Omega }_{\delta }}}{\Delta }_{\delta }f\left( y\right) {G}_{{\Omega }_{\delta }}\left( {\cdot, y}\rig... | Proof Note that \( f - \mathop{\sum }\limits_{{y \in {\Omega }_{\delta }}}{\Delta }_{\delta }f\left( y\right) {G}_{{\Omega }_{\delta }}\left( {\cdot, y}\right) \) is harmonic and vanishes on the boundary. Hence, it equals 0 everywhere. | Yes |
Proposition 4.10. There exists \( C > 0 \) such that for any \( \delta > 0 \) and \( y \in 9{Q}_{\delta } \) , \[ \mathop{\sum }\limits_{{x \in {Q}_{\delta }}}\left| {{\nabla }_{x}{G}_{9{Q}_{\delta }}\left( {x, y}\right) }\right| \leq {C\delta }\mathop{\sum }\limits_{{x \in {Q}_{\delta }}}{G}_{9{Q}_{\delta }}\left( {x,... | Proof In the proof, \( {C}_{1},\ldots ,{C}_{6} \) denote universal constants. First assume \( y \in 9{Q}_{\delta } \smallsetminus 3{Q}_{\delta } \) . Using random walks, one can easily show that there exists \( {C}_{1} > 0 \) such that \[ \frac{1}{{C}_{1}}{G}_{9{Q}_{\delta }}\left( {x, y}\right) \leq {G}_{9{Q}_{\delta ... | Yes |
Proposition 4.18. Any s-holomorphic function \( f : {\Omega }_{\delta }^{\diamond } \rightarrow \mathbb{C} \) is preholomorphic on \( {\Omega }_{\delta }^{\diamond } \) . | Proof Let \( f : {\Omega }_{\delta }^{\diamond } \rightarrow \mathbb{C} \) be a \( s \) -holomorphic function. Let \( v \) be a vertex of \( {\mathbb{L}}_{\delta } \cup {\mathbb{L}}_{\delta }^{ \star } \) (this is the vertex set of the dual of the medial lattice). Assume that \( v \in {\Omega }_{\delta }^{ \star } \), ... | Yes |
Theorem 4.19. Let \( f : {\Omega }_{\delta }^{\diamond } \rightarrow \mathbb{C} \) be an s-holomorphic function on the discrete simply connected domain \( {\Omega }_{\delta }^{\diamond } \), and \( {b}_{0} \in {\Omega }_{\delta } \) . Then, there exists a unique function \( H : {\Omega }_{\delta } \cup {\Omega }_{\delt... | Proof The uniqueness of \( H \) is straightforward since \( {\Omega }_{\delta }^{\diamond } \) is simply connected. To obtain the existence, construct the value at some point by summing increments along an arbitrary path from \( {b}_{0} \) to this point. The only thing to check is that the value obtained does not depen... | Yes |
Proposition 4.20. If \( f : {\Omega }_{\delta }^{\diamond } \rightarrow \mathbb{C} \) is s-holomorphic, then \( {H}^{ \bullet } \) and \( {H}^{\diamond } \) are respectively subharmonic and superharmonic. | Proof Let \( B \) be a vertex of \( {\Omega }_{\delta } \smallsetminus \partial {\Omega }_{\delta } \) . We aim to show that the sum of increments of \( {H}^{ \bullet } \) between \( B \) and its four neighbors is positive. In other words, we need to prove that the sum of increments along the sixteen arrows drawn in Fi... | Yes |
Lemma 5.1. For an edge \( e \in {\Omega }_{\delta }^{\diamond },{F}_{\delta }\left( e\right) \) belongs to \( \ell \left( e\right) \) . | Proof The winding at an edge \( e \) can only take its value in the set \( W + {2\pi }\mathbb{Z} \) where \( W \) is the winding at \( e \) of an arbitrary interface passing through \( e \) . Therefore, the winding weight involved in the definition of \( {F}_{\delta }\left( e\right) \) is always proportional to \( {\ma... | Yes |
Lemma 5.4. The function \( {H}_{\delta }^{ \bullet } \) is equal to 1 on the arc \( {\partial }_{ba} \) . The function \( {H}_{\delta }^{ \circ } \) is equal to 0 on the arc \( {\partial }_{ab}^{ \star } \) . | Proof We first prove that \( {H}_{\delta }^{ \bullet } \) is constant on \( {\partial }_{ba} \) . Let \( B \) and \( {B}^{\prime } \) be two adjacent consecutive sites of \( {\partial }_{ba} \) . They are both adjacent to the same dual vertex \( W \in {\Omega }_{\delta }^{ \star } \), see Fig. 12. Let \( e \) (resp. \(... | Yes |
Lemma 5.5. The function \( {H}_{\delta }^{ \bullet } \) converges to 0 on the arc \( {\partial }_{ab} \) uniformly away from a and \( b,{H}_{\delta }^{ \circ } \) converges to 1 on the arc \( {\partial }_{ba}^{ \star } \) uniformly away from a and \( b \) . | Proof Once again, we prove the result for \( {H}_{\delta }^{ \bullet } \) . The same reasoning then holds for \( {H}_{\delta }^{ \circ } \) . Let \( B \) be a site of \( {\partial }_{ab} \) at distance \( r \) of \( {\partial }_{ba} \) (and therefore at graph distance \( r/\delta \) of \( {\partial }_{ba} \) in \( \lef... | Yes |
Proposition 5.6. Let \( \left( {\Omega, a, b}\right) \) be a simply connected domain with two points on the boundary. Then, \( {\left( {H}_{\delta }\right) }_{\delta > 0} \) converges to \( \operatorname{Im}\left( \phi \right) \) uniformly on any compact subsets of \( \Omega \) when \( \delta \) goes to 0, where \( \ph... | Proof From the definition of \( H,{H}_{\delta }^{ \bullet } \) is subharmonic, let \( {h}_{\delta }^{ \bullet } \) be the preharmonic function with same boundary conditions as \( {H}_{\delta }^{ \bullet } \) on \( \partial {\Omega }_{\delta } \) . Note that \( {H}_{\delta }^{ \bullet } \leq {h}_{\delta }^{ \bullet } \)... | Yes |
Proposition 5.7. For \( \delta > 0,{F}_{\delta } \) is s-holomorphic on \( {\Omega }_{\delta }^{\diamond } \) . | Proof Let \( x, y \) two adjacent medial vertices connected by the edge \( e = \left\lbrack {xy}\right\rbrack \) . Let \( v \) be the vertex of \( {\Omega }_{\delta } \) bordering the (medial) edge \( e \) . As before, set \( {x}_{\omega } \) (resp. \( {y}_{\omega } \) ) for the contribution of \( \omega \) to \( {F}_{... | No |
Lemma 6.3 (Circuits in annuli). Let \( \mathcal{E}\left( {x, n, N}\right) \) be the probability that there exists an open path connecting the boundaries of \( {S}_{n, N}\left( x\right) \) . There exists a constant \( c < 1 \) such that for all \( n > 0 \) , \[ {\phi }_{{p}_{sd},{S}_{n,{2n}}\left( x\right) }^{1}\left( {... | Proof Assume \( x = 0 \) . The result follows from Theorem 3.16 (proved in Section 7.2) applied in the four rectangles \( {R}_{B} = \left\lbrack {-{2n},{2n}}\right\rbrack \times \left\lbrack {-n, - {2n}}\right\rbrack ,\overline{{R}_{L}} = \left\lbrack {-{2n}, - n}\right\rbrack \times \left\lbrack {-{2n},\overline{2n}}\... | Yes |
Lemma 6.6. Let \( \delta > 0 \) . The FK fermionic observable \( {M}_{n}^{\delta }\left( z\right) = {F}_{{\Omega }_{\delta } \smallsetminus \gamma \left\lbrack {0, n}\right\rbrack ,{\gamma }_{n},{b}_{\delta }}\left( z\right) \) is a martingale with respect to \( \left( {\mathcal{F}}_{n}\right) \), where \( {\mathcal{F}... | Proof For a Dobrushin domain \( \left( {{\Omega }_{\delta }^{\diamond },{a}_{\delta },{b}_{\delta }}\right) \), the slit domain created by \ | No |
Lemma 7.1. Let \( p \in \left( {0,1}\right) \) . Consider a vertex \( v \in {\Omega }^{\diamond } \smallsetminus \partial {\Omega }^{\diamond } \) ,\n\n\[ F\left( A\right) - F\left( C\right) = i{\mathrm{e}}^{\mathrm{i}\alpha }\left\lbrack {F\left( B\right) - F\left( D\right) }\right\rbrack \]\n\n(7.1)\n\nwhere \( A \) ... | The proof of this statement follows along the same lines as the proof of Lemma 5.2 | No |
Proposition 7.2. For \( p < \sqrt{2}/\left( {1 + \sqrt{2}}\right) \), there exists \( \xi = \xi \left( p\right) > 0 \) such that for every \( n \) ,\n\n\[{\phi }_{p}\left( {0 \leftrightarrow \mathrm{i}n}\right) \leq {\mathrm{e}}^{-{\xi n}}\]\n\nwhere the mesh size of the lattice \( \mathbb{L} \) is 1 . | In this proof, the lattices are rotated by an angle \( \pi /4 \) . We will be able to estimate the connectivity probabilities using the FK fermionic observable. Indeed, the observable on the free boundary is related to the probability that sites are connected to the wired arc. More precisely: | No |
Lemma 7.3. Fix \( \left( {G, a, b}\right) \) a Dobrushin domain and \( p \in \left( {0,1}\right) \) . Let \( u \in G \) be a site on the free arc, and e be a side of the black diamond associated to \( u \) which borders a white diamond of the free arc. Then,\n\n\[ \left| {F\left( e\right) }\right| = {\phi }_{p, G}^{a, ... | Proof Let \( u \) be a site of the free arc and recall that the exploration path is the interface between the open cluster connected to the wired arc and the dual open cluster connected to the free arc. Since \( u \) belongs to the free arc, \( u \) is connected to the wired arc if and only if \( e \) is on the explora... | Yes |
Theorem 7.4. The critical parameter for the FK-Ising model is \( \sqrt{2}/\left( {1 + \sqrt{2}}\right) \). The critical inverse-temperature for the Ising model is \( \frac{1}{2}\ln \left( {1 + \sqrt{2}}\right) \). | Proof The inequality \( {p}_{c} \geq \sqrt{2}/\left( {1 + \sqrt{2}}\right) \) follows from Proposition 7.2 since there is no infinite cluster for \( {\phi }_{p,2}^{0} \) when \( p < {p}_{sd} \) (the probability that 0 and \( \overline{in} \) are connected converges to 0 ). In order to prove that \( {p}_{c} \leq \sqrt{2... | Yes |
Proposition 7.5. Let \( p \neq {p}_{sd} \), \[ {\Delta }_{\delta }{F}_{\delta }\left( v\right) = \left( {\cos {2\alpha } - 1}\right) {F}_{\delta }\left( v\right) \] for every \( v \in {\Omega }_{\delta }^{\diamond } \smallsetminus \partial {\Omega }_{\delta }^{\diamond } \), where \( {\Delta }_{\delta } \) is the avera... | When \( \delta \) goes to 0, one can perform two scaling limits. If \( p = {p}_{sd}\left( {1 - {\lambda \delta }}\right) \) goes to \( {p}_{sd} \) as \( \delta \) goes to \( 0,\frac{1}{{\delta }^{2}}\left( {{\Delta }_{\delta } + \left\lbrack {1 - \cos {2\alpha }}\right\rbrack I}\right) \) converges to \( \Delta + {\lam... | Yes |
Theorem 7.6. Fix \( \beta < {\beta }_{c} \) (and \( \alpha \) associated to it) and set\n\n\[ m\left( \beta \right) \mathrel{\text{:=}} \cos \left( {2\alpha }\right) . \]\n\nFor any \( x \in \mathbb{L} \) ,\n\n\[ - \mathop{\lim }\limits_{{n \rightarrow \infty }}\frac{1}{n}\ln {\mu }_{\beta }\left\lbrack {{\sigma }_{0}\... | The massive Green function \( {G}_{m}\left( {0, x}\right) \) on the right of (7.9) has been widely studied. In particular, we can compute the rate of decay in any direction and deduce Theorem 2.3 and Theorem 2.4 (see e.g. [Mes06]). | No |
Proposition 7.9. There exist constants \( 0 < c, C,\delta ,\Delta < \infty \) such that for any sites \( x, y \in \mathbb{L} \) ,\n\n\[ \n\frac{c}{{\left| x - y\right| }^{\delta }} \leq {\mu }_{{\beta }_{c}}\left\lbrack {{\sigma }_{x}{\sigma }_{y}}\right\rbrack \leq \frac{C}{{\left| x - y\right| }^{\Delta }} \n\]\n\nwh... | Proof Using the Edwards-Sokal coupling, 7.13 can be rephrased as\n\n\[ \n\frac{c}{{\left| x - y\right| }^{\delta }} \leq {\phi }_{{p}_{sd},2}\left\lbrack {x \leftrightarrow y}\right\rbrack \leq \frac{C}{{\left| x - y\right| }^{\Delta }}, \]\n\nwhere \( {\phi }_{{p}_{sd},2} \) is the unique FK-Ising infinite-volume meas... | No |
Lemma 7.11. Let \( \left\lbrack {xy}\right\rbrack \) be an horizontal edge of \( {\Omega }_{\delta } \) . Then\n\n\[ \n\lambda {\mu }_{{\beta }_{c},{\Omega }_{\delta }}^{f}\left\lbrack {{\sigma }_{x}{\sigma }_{y}}\right\rbrack = {P}_{\ell \left( {ac}\right) }\left\lbrack {{F}_{{\Omega }_{\delta }}^{a}\left( c\right) }\... | Proof of Theorem 2.12 (Sketch) The function \( {F}_{\delta }^{{a}_{\delta }}/\delta \) converges uniformly on any compact subset of \( \Omega \smallsetminus \{ a\} \) . This fact is not helpful, since the interesting values of \( {F}_{\delta }^{{a}_{\delta }} \) are located at neighbors of the singularity. It can be pr... | No |
Proposition 8.6. For \( q \leq 4 \) and any FK Dobrushin domain, consider the observable \( F \) at criticality with spin \( \sigma = 1 - \frac{2}{\pi }\arccos \left( {\sqrt{q}/2}\right) \) . For any medial vertex inside the domain, \[ F\left( N\right) - F\left( S\right) = i\left\lbrack {F\left( E\right) - F\left( W\ri... | These relations can be understood as Cauchy-Riemann equations around some vertices. Importantly, \( F \) is not determined by these relations for general \( q \) (the number of variables exceeds the number of equations). For \( q = 2 \), which corresponds to \( \sigma = 1/2 \), the complex argument modulo \( \pi \) of ... | Yes |
Theorem 1.1.1 For any two distinct points \( p \) and \( q \) there exists a point \( r \) which lies between \( p \) and \( q \), i.e. the line segment \( \overline{pq} \) is not empty. | Proof Let \( p \) and \( q \) be two points. By axiom \( {\mathrm{I}}_{4} \) there exists a point \( s \) that does not lie on the straight line \( L\left( {p, q}\right) \) . By axiom \( {\mathrm{A}}_{3} \) there is a point \( t \) such that \( s \) lies between \( p \) and \( t \) . Another application of axiom \( {\m... | Yes |
Lemma1.1.4 The congruence of line segments defines an equivalence relation on the set of line segments. | Proof (a) Let \( \overline{pq} \) be a line segment. We show that \( \overline{pq} \) is congruent to itself. Let \( L \) be a straight line that contains \( p, p \in L \) . By axiom \( {\mathrm{K}}_{1} \) there exists a point \( r \) on \( L \) such that \( \overline{pq} \) and \( \overline{pr} \) are congruent. Then ... | Yes |
Theorem 1.1.6 Let \( \left( {p, q, r}\right) \) be a triple of points that do not lie on a straight line, \( \left( {{p}_{1},{q}_{1},{r}_{1}}\right) \) likewise. If \( \overline{pq} \equiv \overline{{p}_{1}{q}_{1}},\overline{pr} \equiv \overline{{p}_{1}{r}_{1}} \) and \( \angle \left( {q, p, r}\right) \equiv \angle \le... | Proof The angle congruences follow directly from axiom \( {\mathrm{K}}_{6} \), in the second case after renaming the variables. It remains to show that \( \overline{qr} \equiv \overline{{q}_{1}{r}_{1}} \) . By axiom \( {\mathrm{K}}_{1} \) we can find a point \( {s}_{1} \) on the straight line \( L\left( {{q}_{1},{r}_{1... | No |
Theorem 1.1.7 (Congruence of adjacent angles) Suppose that the pairwise distinct points \( p, q \) and \( s \) lie on a straight line \( L \), while \( r \notin L \) . Analogously, let \( {p}_{1},{q}_{1},{s}_{1} \in {L}_{1} \) be pairwise distinct, \( {r}_{1} \notin {L}_{1} \) . If \( \angle \left( {p, q, r}\right) \) ... | Proof The points \( {p}_{1},{r}_{1} \) and \( {s}_{1} \) can by axiom \( {\mathrm{K}}_{1} \) be assumed to have been chosen in such a way that \( \overline{pq} \equiv \overline{{p}_{1}{q}_{1}},\overline{rq} \equiv \overline{{r}_{1}{q}_{1}} \) and \( \overline{sq} \equiv \overline{{s}_{1}{q}_{1}} \) . From theorem 1.1.6... | Yes |
Theorem 1.1.8 (Congruence of vertical angles) Let \( L \) and \( M \) be two distinct straight lines that intersect at \( p \) . Let \( r, q \in L \) lie on two distinct sides of \( p \) and let \( s, t \in M \) lie on two distinct sides of \( p \) as well. Then\n\n\[ \angle \left( {q, p, s}\right) \equiv \angle \left(... | Proof Both \( \angle \left( {q, p, s}\right) \) and \( \angle \left( {r, p, t}\right) \), are adjacent angles of \( \angle \left( {q, p, t}\right) \) . The angle \( \angle \left( {q, p, t}\right) \) is congruent to itself by axiom \( {\mathrm{K}}_{4} \) . The claim therefore follows by theorem 1.1.7. | Yes |
Theorem 1.1.9 (Existence of a parallel) Let \( L \) be a straight line, \( p \) a point, \( p \notin L \). There then exists a straight line \( M \) that contains \( p \) and that does not intersect \( L \). | Proof Let \( L \) be a straight line and \( p \) a point that does not lie on \( L \). We will first construct the line \( M \) and then show that it has the desired properties.\n\nFor the construction we choose a point \( q \in L \) and add the straight line \( N \mathrel{\text{:=}} \) \( L\left( {p, q}\right) \). We ... | Yes |
Theorem 1.2.4 (cosine rule for Euclidean geometry) Let \( p, q, r \in {\mathbb{R}}^{2} \) . Let \( a = \) \( \parallel p - q\parallel, b = \parallel p - r\parallel \) and \( c = \parallel q - r\parallel \) be the sides of a triangle with vertices \( p, q \) and \( r \) . Let \( \gamma \) be the interior angle at the ve... | Proof Euclidean motions do not change the side and angle ratios. We can therefore after the application of a suitable Euclidean motion assume that \( p = {\left( 0,0\right) }^{\top }, q = {\left( a,0\right) }^{\top } \) and \( r = {\left( x, y\right) }^{\top } \) .\n\n Using the notation from theorem 1.2.4 and letting the angle be a right angle, \( \gamma = \pi /2 \), the following equality holds: | \[ {a}^{2} + {b}^{2} = {c}^{2} \] | Yes |
Theorem 1.2.6 (sine rule for Euclidean geometry) If \( \beta \) denotes the interior angle at vertex \( q \) and \( \alpha \) the one at vertex \( r \), then the following equality holds:\n\n\[ \frac{a}{b} = \frac{\sin \left( \alpha \right) }{\sin \left( \beta \right) } \] | Proof By the cosine rule the angle at vertex \( r \) satisfies\n\n\[ - {2bc}\cos \left( \alpha \right) = {a}^{2} - \left( {{b}^{2} + {c}^{2}}\right) \]\n\nand hence\n\n\[ 4{b}^{2}{c}^{2}{\cos }^{2}\left( \alpha \right) = {\left( -{a}^{2} + {b}^{2} + {c}^{2}\right) }^{2}. \]\n\nAnalogously,\n\n\[ 4{a}^{2}{c}^{2}{\cos }^... | Yes |
Theorem 1.2.7 (angle sum in the Euclidean triangle) The sum of the interior angles in the Euclidean triangle satisfies\n\n\\[\n\\alpha + \\beta + \\gamma = \\pi \\text{.}\n\\] | Proof (a) We first prove the statement for right-angle triangles. Let \\( \\gamma = \\) \\( \\pi /2 \\) . Then\n\n\\[\n\\sin \\left( {\\alpha + \\beta + \\pi /2}\\right) = \\cos \\left( {\\alpha + \\beta }\\right) = \\cos \\left( \\alpha \\right) \\cos \\left( \\beta \\right) - \\sin \\left( \\alpha \\right) \\sin \\le... | Yes |
Example 2.1.2 A straight line can be described as a regular parametrised curve: | \[ c : \mathbb{R} \rightarrow {\mathbb{R}}^{n}, \] \[ c\left( t\right) = {c}_{0} + t \cdot v \] where \( {c}_{0} \in {\mathbb{R}}^{n} \) and \( v \in {\mathbb{R}}^{n} - \{ 0\} \) . This obviously satisfies the condition \( \dot{c}\left( t\right) = \) \( v \neq 0 \) . | Yes |
Example 2.1.3 A circular curve in the plane around the origin \( \\left( {0,0}\\right) \) with radius \( r > 0 \) looks as follows:\n\n\[ c : \\mathbb{R} \\rightarrow {\\mathbb{R}}^{2} \]\n\n\[ c\\left( t\\right) = \\left( \\begin{matrix} r \\cdot \\cos \\left( t\\right) \\\\ r \\cdot \\sin \\left( t\\right) \\end{matr... | The arrow in the sketch shows the direction in which the curve traverses the image. This example shows that a regular parametrised curve is not necessarily injective. Because of the periodicity \( c\\left( {t + {2\\pi }}\\right) = c\\left( t\\right) \) the curve runs infinitely often through every point that is in the ... | No |
Proposition 2.1.13 For every regular parametrised curve \( c \) there exists an orientation-preserving parameter transformation \( \varphi \) such that the reparametrisation \( c \circ \varphi \) is parametrised by arc-length. | Proof Let \( c : I \rightarrow {\mathbb{R}}^{n} \) be a regular parametrised curve. We choose \( {t}_{0} \in I \) and set \[ \psi \left( s\right) \mathrel{\text{:=}} {\int }_{{t}_{0}}^{s}\parallel \dot{c}\left( t\right) \parallel {dt} \] Since \( {\psi }^{\prime }\left( s\right) = \parallel \dot{c}\left( s\right) \para... | Yes |
Lemma 2.1.14 Let \( {c}_{1} : {I}_{1} \rightarrow {\mathbb{R}}^{n} \) and \( {c}_{2} : {I}_{2} \rightarrow {\mathbb{R}}^{n} \) be different parametri-sations by arc-length of the same curve, then the corresponding parameter transformation \( \varphi : {I}_{1} \rightarrow {I}_{2} \) with \( {c}_{1} = {c}_{2} \circ \varp... | Proof We have\n\n\[ 1 = \begin{Vmatrix}{{\dot{c}}_{1}\left( t\right) }\end{Vmatrix} = \begin{Vmatrix}{{\dot{c}}_{2}\left( {\varphi \left( t\right) }\right) \cdot \dot{\varphi }\left( t\right) }\end{Vmatrix} = \begin{Vmatrix}{{\dot{c}}_{2}\left( {\varphi \left( t\right) }\right) }\end{Vmatrix} \cdot \left| {\dot{\varphi... | Yes |
Lemma 2.1.16 The length of a parametrised curve is not changed by reparametrisation. | Proof This can be deduced using the substitution rule. If \( \widetilde{c} = c \circ \varphi \) is a reparametrisation of \( c,\varphi : \left\lbrack {{a}^{\prime },{b}^{\prime }}\right\rbrack \rightarrow \left\lbrack {a, b}\right\rbrack \), then we have\n\n\[ L\left\lbrack \widetilde{c}\right\rbrack = {\int }_{{a}^{\p... | Yes |
Proposition 2.1.18 (Approximation of length with polygons) Let \( c : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{n} \) be a parametrised curve. Then for every (however small) \( \varepsilon > 0 \) there exists a \( \delta > 0 \) such that for every partition \( a = {t}_{0} < {t}_{1} < \cdots < {t}_{k} =... | Proof Let \( \varepsilon > 0 \) be given. We choose an \( {\varepsilon }^{\prime } \in \left( {0,\varepsilon /\left( {1 + \sqrt{n}\left( {b - a}\right) }\right) }\right) \) . According to the theorem about Riemann sums [18, p. 106, theorem 2.7] applied to the integral\n\n\[ L\left\lbrack c\right\rbrack = {\int }_{a}^{b... | Yes |
Example 2.2.3 Let us consider the circular line \( c : \mathbb{R} \rightarrow {\mathbb{R}}^{2} \) of radius \( r > 0 \), parametrised by arc-length \( c\left( t\right) = {\left( r \cdot \cos \left( t/r\right), r \cdot \sin \left( t/r\right) \right) }^{\top } \) . Then \( \dot{c}\left( t\right) = \) | \( {\left( -\sin \left( t/r\right) ,\cos \left( t/r\right) \right) }^{\top } \) and \( \ddot{c}\left( t\right) = \left( {1/r}\right) {\left( -\cos \left( t/r\right) , - \sin \left( t/r\right) \right) }^{\top } = \left( {1/r}\right) n\left( t\right) \) . Hence \( \kappa \equiv 1/r \) . | Yes |
Proposition 2.2.4 (Frenet formulae) Let \( c : I \rightarrow {\mathbb{R}}^{2} \) be a plane unit speed curve. We set \( v \mathrel{\text{:=}} \dot{c} \) . Let \( \kappa \) be the curvature of \( c \) and let \( n \) be the normal vector. Then\n\n\[ \left( {\dot{v}\left( t\right) ,\dot{n}\left( t\right) }\right) = \left... | Proof The equation \( \dot{v} = \kappa \cdot n \) is exactly the definition of the curvature. By differentiating the equation \( \langle n, n\rangle \equiv 1 \) we conclude as above that \( \dot{n}\left( t\right) \) is perpendicular to \( n\left( t\right) \) and hence must be a multiple of \( v\left( t\right) ,\dot{n}\... | Yes |
Lemma 2.2.5 Let \( c : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbb{R}}^{2} \) be a unit speed curve. Then there exists a \( {C}^{\infty } \) -function \( \vartheta : \left\lbrack {a, b}\right\rbrack \rightarrow \mathbb{R} \) such that\n\n\[ \dot{c}\left( t\right) = \left( \begin{matrix} \cos \left( {\vartheta... | Proof (a) We first consider the case that the image \( \dot{c}\left( \left\lbrack {a, b}\right\rbrack \right) \) is fully contained in one of the following four semicircles:\n\n\[ {S}_{R} \mathrel{\text{:=}} \{ {\left( x, y\right) }^{\top } \in {S}^{1} \subset {\mathbb{R}}^{2}\left| {x > 0\} ,\;{S}_{L} \mathrel{\text{:... | Yes |
The circle of radius \( r > 0 \) has the parametrisation by arclength \( c\left( t\right) = {\left( r \cdot \cos \left( t/r\right), r \cdot \sin \left( t/r\right) \right) }^{\top } \) with period \( L = {2\pi r} \) . For the velocity vector we obtain | \[ \dot{c}\left( t\right) = \left( \begin{matrix} - \sin \left( {t/r}\right) \\ \cos \left( {t/r}\right) \end{matrix}\right) \] \[ = \left( \begin{matrix} \cos \left( {t/r + \pi /2}\right) \\ \sin \left( {t/r + \pi /2}\right) \end{matrix}\right) \] Hence we obtain the trigonometric function \( \vartheta \left( t\right)... | Yes |
Theorem 2.2.9 Let \( c : \mathbb{R} \rightarrow {\mathbb{R}}^{2} \) be a plane unit speed curve with period \( L \) . Let \( \kappa : \mathbb{R} \rightarrow \mathbb{R} \) be the curvature of \( c \) . Then\n\n\[ \n{n}_{c} = \frac{1}{2\pi }{\int }_{0}^{L}\kappa \left( t\right) {dt} \n\] | Proof We write as in lemma \( {2.2.5}\dot{c}\left( t\right) = {\left( \cos \left( \vartheta \left( t\right) \right) ,\sin \left( \vartheta \left( t\right) \right) \right) }^{\top } \) . Differentiation gives \( \ddot{c}\left( t\right) = {\left( -\sin \left( \vartheta \left( t\right) \right) \cdot \dot{\vartheta }\left(... | Yes |
Theorem 2.2.10 (Hopf’s Umlaufsatz) \( {}^{2}\; \) A simple closed plane curve has winding number 1 or -1 . | To prove Hopf's Umlaufsatz we first need a generalisation of lemma 2.2.5. | No |
Example 2.2.17 Let us consider the ellipse, parametrised by\n\n\[ c : \mathbb{R} \rightarrow {\mathbb{R}}^{2},\;c\left( t\right) = \left( \begin{array}{l} a\cos \left( t\right) \\ b\sin \left( t\right) \end{array}\right) ,\n\]\n\nwith \( 0 < a < b \) . | This is unfortunately not a parametrisation by arc-length. Instead of trying to reparametrise the ellipse by arc-length, we use the formula for curvature from exercise 2.10:\n\n\[ \kappa \left( t\right) = \frac{\det \left( {\dot{c}\left( t\right) ,\ddot{c}\left( t\right) }\right) }{\parallel \dot{c}\left( t\right) {\pa... | Yes |
Theorem 2.2.18 (Four-vertex theorem) If \( c : \mathbb{R} \rightarrow {\mathbb{R}}^{2} \) is a convex periodic plane curve parametrised by arc-length and with period \( L \), then \( c \) has at least four vertices in \( \lbrack 0, L) \) . | To prove the above theorem we need the following lemmas. | No |
Lemma 2.2.20 If a simple closed convex plane curve intersects a straight line at more than one point tangentially, then the curve contains a segment of some straight line. | Proof of the lemma If the curve has more than two points of intersection with the straight line \( G \), then the claim follows from lemma 2.2.19. Hence we can assume that the curve intersects the straight line \( G \) at exactly two points. Because of the convexity of the curve it must lie entirely on one side of the ... | Yes |
Theorem 2.2.21 (isoperimetric inequality) Let \( G \subset {\mathbb{R}}^{2} \) be a bounded region with the simple closed plain curve \( c \) as boundary. Let \( A\left\lbrack G\right\rbrack \) be the area of the surface. Then\n\n\[ \n{4\pi A}\left\lbrack G\right\rbrack \leq L{\left\lbrack c\right\rbrack }^{2} \n\]\n\n... | To prove the isoperimetric inequality we first need a lemma which tells us how to find the surface area of \( G \) using the boundary curve \( c \). | No |
Proposition 2.3.7 (Frenet formulae) Let \( c : I \rightarrow {\mathbb{R}}^{3} \) be a space curve parametrised by arc-length with positive curvature, \( \kappa \left( t\right) > 0 \) for all \( t \in I \) . Let \( \left( {v, n, b}\right) \) be the Frenet dreibein of \( c \), and let \( \tau \) be the torsion. Then\n\n\... | Proof The equation \( \dot{v} = \kappa \cdot n \) is exactly the definition of the normal vector. Thus the first column of the \( 3 \times 3 \) -matrix is correct.\n\nThe second column follows from \( \langle \dot{n}, v\rangle = \left( {d/{dt}}\right) \langle n, v\rangle - \langle n,\dot{v}\rangle = 0 - \kappa = - \kap... | Yes |
Example 2.3.11 Let us consider a triangle \( P = \left( {{a}_{1},{a}_{2},{a}_{3}}\right) \) . The exterior angle \( {\alpha }_{i} \) and the interior angle \( {\beta }_{i} \) at a vertex \( {a}_{i} \) always add up to \( \pi \), i.e. \( {\alpha }_{i} + {\beta }_{i} = \pi \) . According to theorem 1.2.7 the sum of the i... | \[ \kappa \left( P\right) = {\alpha }_{1} + {\alpha }_{2} + {\alpha }_{3} \] \[ = {\alpha }_{1} + {\beta }_{1} + {\alpha }_{2} + {\beta }_{2} + {\alpha }_{3} + {\beta }_{3} - \left( {{\beta }_{1} + {\beta }_{2} + {\beta }_{3}}\right) \] \[ = {3\pi } - \pi = {2\pi }\text{.} \] | Yes |
Lemma 2.3.15 Let \( c : \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \rightarrow {\mathbb{R}}^{3} \) be a segment of a curve parametrised by arc-length with \( \parallel \ddot{c}\left( u\right) - \ddot{c}\left( v\right) \parallel < \varepsilon \) for all \( u, v \in \left\lbrack {{t}_{0},{t}_{1}}\right\rbrack \) . Let \... | Proof We compute\n\n\[ {\int }_{\tau }^{{t}_{1}}\left( {{\int }_{\tau }^{v}\left( {\ddot{c}\left( u\right) - \ddot{c}\left( \tau \right) }\right) {du}}\right) {dv} \]\n\n\[ = {\int }_{\tau }^{{t}_{1}}\left( {\dot{c}\left( v\right) - \dot{c}\left( \tau \right) - \left( {v - \tau }\right) \ddot{c}\left( \tau \right) }\ri... | Yes |
Proposition 2.3.17 Let \( c \) be a closed space curve. Then\n\n\[ \n\frac{1}{A\left\lbrack {S}^{2}\right\rbrack }{\int }_{{S}^{2}}\mu \left( {c, e}\right) {dA}\left( e\right) = \frac{\kappa \left( c\right) }{2\pi }.\n\]\nHere \( A\left\lbrack {S}^{2}\right\rbrack = {4\pi } \) denotes the surface area of the two-dimens... | Proof (a) We prove the claim first for polygons. Let \( P = \left( {{a}_{1},\ldots ,{a}_{m}}\right) \) be a closed polygon. We set \( {b}_{j} \mathrel{\text{:=}} \left( {{a}_{j} - {a}_{j - 1}}\right) /\begin{Vmatrix}{{a}_{j} - {a}_{j - 1}}\end{Vmatrix} \in {S}^{2} \) . For \( e \in {S}^{2} \) we define a circular line ... | Yes |
Corollary 2.3.18 Let \( c \) be a closed space curve. Then\n\n\[ \kappa \left( c\right) \geq {2\pi \mu }\left( c\right) \] | Proof \( \;\kappa \left( c\right) /{2\pi } \) is the mean of the function \( \mu \left( {c, \cdot }\right) \) by proposition 2.3.17, while \( \mu \left( c\right) \) is the minimum. | Yes |
Theorem 2.3.19 (Fenchel's theorem) Let \( c \) be a simple closed space curve. Then\n\n\[ \kappa \left( c\right) \geq {2\pi }\text{.} \]\n\nWe have the equality \( \kappa \left( c\right) = {2\pi } \) exactly if \( c \) is a convex plane curve. | Proof (a) For every \( e \in {S}^{2} \) the function \( t \mapsto \langle c\left( t\right), e\rangle \) has at least one maximum and one minimum. Thus \( \mu \left( c\right) \geq 1 \) . By corollary 2.3.18 it follows that\n\n\[ \kappa \left( c\right) \geq {2\pi \mu }\left( c\right) \geq {2\pi } \]\n\n(b) Now let \( c \... | Yes |
Example 3.1.3 (affine planes) The simplest examples of regular surfaces are affine planes. The affine plane through a point \( p \in {\mathbb{R}}^{3} \), spanned by the linearly independent vectors \( X, Y \in {\mathbb{R}}^{3} \), is the set\n\n\[ S = \left\{ {p + {u}^{1} \cdot X + {u}^{2} \cdot Y \mid {u}^{1},{u}^{2} ... | We can use a single parametrisation. We set \( V \mathrel{\text{:=}} {\mathbb{R}}^{3}, U \mathrel{\text{:=}} {\mathbb{R}}^{2} \) and \( F : U \rightarrow {\mathbb{R}}^{3} \) , \( F\left( {{u}^{1},{u}^{2}}\right) \mathrel{\text{:=}} p + {u}^{1} \cdot X + {u}^{2} \cdot Y \) . | Yes |
Example 3.1.4 (graph of a function) Let \( U \subset {\mathbb{R}}^{2} \) be open, \( f : U \rightarrow \mathbb{R} \) a smooth function. We consider the graph of \( f \) , \[ S = \left\{ {{\left( x, y, z\right) }^{\top } \in {\mathbb{R}}^{3}\left| {\;{\left( x, y\right) }^{\top } \in U}\right., z = f\left( {x, y}\right)... | In this case we can also use a single coordinate neighbourhood. Again we set \( V \mathrel{\text{:=}} {\mathbb{R}}^{3} \) and \[ F : U \rightarrow {\mathbb{R}}^{3},\;F\left( {x, y}\right) \mathrel{\text{:=}} {\left( x, y, f\left( x, y\right) \right) }^{\top }. \] Then obviously \( F\left( U\right) = S = S \cap V \) . F... | Yes |
Example 3.1.5 (the sphere) We consider\n\n\[ S = {S}^{2} = \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3}\\left| {\\;{x}^{2} + {y}^{2} + {z}^{2} = 1}\\right. }\\right\\} .\n\] | Let us first set \( V \\mathrel{\\text{:=}} {V}_{3}^{ + } \\mathrel{\\text{:=}} \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3} \\mid z > 0}\\right\\} \) . Then \( {S}^{2} \\cap {V}_{3}^{ + } \) is the graph of the function \( f\\left( {x, y}\\right) = \\sqrt{1 - \\left( {{x}^{2} + {y}^{2}}\\right... | Yes |
Proposition 3.1.6 Let \( {V}_{0} \subset {\mathbb{R}}^{3} \) be open, let \( f : {V}_{0} \rightarrow \mathbb{R} \) be a smooth function.\n\nWe set \( S \mathrel{\text{:=}} \left\{ {{\left( x, y, z\right) }^{\top } \in V \mid f\left( {x, y, z}\right) = 0}\right\} \) . If\n\n\[ \operatorname{grad}f\left( p\right) \neq {\... | Proof Let \( p \mathrel{\text{:=}} {\left( {x}_{0},{y}_{0},{z}_{0}\right) }^{\top } \in S \) . Since\n\n\[ \operatorname{grad}f\left( p\right) = {\left( \frac{\partial f}{\partial x}\left( p\right) ,\frac{\partial f}{\partial y}\left( p\right) ,\frac{\partial f}{\partial z}\left( p\right) \right) }^{\top } \neq {\left(... | Yes |
Example 3.1.7 (Ellipsoid) Let us consider\n\n\[ S \mathrel{\text{:=}} \left\{ {{\left( x, y, z\right) }^{\top } \in {\mathbb{R}}^{3}\left| {\;\frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} + \frac{{z}^{2}}{{c}^{2}} = 1}\right. }\right\} \]\n\nfor non-vanishing constants \( a, b, c \in \mathbb{R} \) . | If we set\n\n\[ {V}_{0} \mathrel{\text{:=}} {\mathbb{R}}^{3},\;f : {\mathbb{R}}^{3} \rightarrow \mathbb{R},\;f\left( {x, y, z}\right) \mathrel{\text{:=}} \frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} + \frac{{z}^{2}}{{c}^{2}} - 1, \]\n\nthen\n\n\[ S = \left\{ {{\left( x, y, z\right) }^{\top } \in {\mathbb{R}}^{3} \... | Yes |
We now want to think about whether the double cone\n\n\[ S = \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3}\\left| {\\;{x}^{2} + {y}^{2} = {z}^{2}}\\right. }\\right\\} \]\n\nis a regular surface or not. | Suppose that \( S \) were a regular surface. Then a local parametrisation around \( p = {\\left( 0,0,0\\right) }^{\\top } \) would exist, i.e. there would exist open subsets \( V \\subset {\\mathbb{R}}^{3}, U \\subset {\\mathbb{R}}^{2} \) and a smooth map \( F : U \\rightarrow V \), such that \( F\\left( U\\right) = S ... | Yes |
Corollary 3.1.10 Let \( S \) be a regular surface with local parametrisations \( \left( {{U}_{1},{F}_{1},{V}_{1}}\right) \) and \( \left( {{U}_{2},{F}_{2},{V}_{2}}\right) \) . Then\n\n\[ \n{F}_{2}^{-1} \circ {F}_{1} : {F}_{1}^{-1}\left( {{V}_{1} \cap {V}_{2}}\right) \rightarrow {F}_{2}^{-1}\left( {{V}_{1} \cap {V}_{2}}... | Proof This follows from the application of proposition 3.1.9 to \n\n\[ \nW = {F}_{1}^{-1}\left( {{V}_{1} \cap {V}_{2}}\right) ,\;\varphi = {F}_{1}\text{ and }\left( {U, F, V}\right) = \left( {{U}_{2},{F}_{2},{V}_{2}}\right) . \n\] | Yes |
Proposition 3.1.11 Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface, \( p \in S \), and \( f : S \rightarrow {\mathbb{R}}^{n} \) a continuous map. Then the following are equivalent:\n\n(1) There is an open neighbourhood \( V \) of \( p \) in \( {\mathbb{R}}^{3} \) and an extension \( \widetilde{f} \) of \( {\l... | Proof\n\n(a) (1) implies (3): \( F \) is smooth and \( \widetilde{f} \) is smooth around \( p \), hence\n\n\[ f \circ F = \widetilde{f} \circ F \]\n\nis also a smooth map on a neighbourhood of \( {F}^{-1}\left( p\right) \) .\n\n(b) (3) implies (2) trivially.\n\n(c) (2) implies (1): we consider the local diffeomorphism\... | Yes |
Example 3.1.14 Let \( A \) be an orthogonal \( 3 \times 3 \) matrix. Being a linear map \( A : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) is certainly \( {C}^{\infty } \). Because of the orthogonality, \( A \) maps the unit sphere to itself. By exercise 3.4, | \[ f = {\left. A\right| }_{{S}^{2}} : {S}^{2} \rightarrow {S}^{2} \] is a smooth map. | No |
Example 3.1.16 Let\n\n\[ \n{S}_{1} = \left\{ {{\left( x, y, z\right) }^{\top } \in {\mathbb{R}}^{3}\left| {\;\frac{{x}^{2}}{{a}^{2}} + \frac{{y}^{2}}{{b}^{2}} + \frac{{t}^{2}}{{c}^{2}} = 1}\right. }\right\}, a, b, c > 0, \n\]\n\nan ellipsoid. Let \( {S}_{2} = {S}^{2} \) be the sphere. Then \( {S}_{1} \) and \( {S}_{2} ... | We may take\n\n\[ \nf : {S}_{1} \rightarrow {S}_{2} \n\]\n\n\[ \nf\left( {x, y, z}\right) = {\left( \frac{x}{a},\frac{y}{b},\frac{z}{c}\right) }^{\top }, \n\]\n\nas the diffeomorphism, for example. | Yes |
Example 3.1.17 Let \( {S}_{1} = \left\{ {{\left( x, y,\varphi \left( x, y\right) \right) }^{\top } \mid {\left( x, y\right) }^{\top } \in U}\right\} \) be the graph of a \( {C}^{\infty } \) -function \( \varphi : U \rightarrow \mathbb{R} \) . Let \( {S}_{2} = U \times \{ 0\} \subset {\mathbb{R}}^{3} \) be \( U \) inter... | \[ f : {S}_{1} \rightarrow {S}_{2},\;f\left( {x, y, z}\right) = {\left( x, y,0\right) }^{\top }, \] \[ {f}^{-1} : {S}_{2} \rightarrow {S}_{1},\;f\left( {x, y,0}\right) = {\left( x, y,\varphi \left( x, y\right) \right) }^{\top }.\] | Yes |
Proposition 3.2.2 Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface, let \( p \in S \) . Further, let \( \left( {U, F, V}\right) \) be a local parametrisation of \( S \) around \( p \) . We set \( {u}_{0} \mathrel{\text{:=}} {F}^{-1}\left( p\right) \in U \) .\n\nThen\n\n\[ \n{T}_{p}S = \operatorname{Image}\left... | Proof (a) We show the inclusion \ | No |
Proposition 3.2.4 Let \( V \subset {\mathbb{R}}^{3} \) be open, let \( f : V \rightarrow \mathbb{R} \) be a smooth function and let \( S = {f}^{-1}\left( 0\right) \subset {\mathbb{R}}^{3} \) . Suppose that \( \operatorname{grad}f\left( p\right) \neq 0 \) for all \( p \in S \) . Then for \( p \in S \) the gradient of \(... | Proof Let \( X \in {T}_{p}S \) . We choose a smooth parametrised curve \( c : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow S \) with \( c\left( 0\right) = p \) and \( \dot{c}\left( 0\right) = X \) . As \( c \) is completely contained in \( S \) we have \( \left( {f \circ c}\right) \left( t\right) = 0 \) for a... | Yes |
The sphere is described by\n\n\[ \n{S}^{2} = {f}^{-1}\left( 0\right) \n\]\n\nwhere \( f\left( {x, y, z}\right) = {x}^{2} + {y}^{2} + {z}^{2} - 1 \) . | We calculate\n\n\[ \n\operatorname{grad}f\left( {x, y, z}\right) = 2\left( {x, y, z}\right) .\n\]\n\nThe tangent plane \( {T}_{p}{S}^{2} \) is hence exactly the orthogonal complement of the root point vector \( p \) . | Yes |
Proposition 3.2.7 This definition makes \( {d}_{p}f \) well-defined, i.e. \( {d}_{p}f\left( X\right) \) depends only on \( X \), not on the particular choice of the curve \( c \) . Further \( {d}_{p}f \) is linear. | Proof We express \( {d}_{p}f \) using local parametrisations. Let \( \left( {{U}_{1},{F}_{1},{V}_{1}}\right) \) be a local parametrisation of \( {S}_{1} \) around \( p \) and let \( \left( {{U}_{2},{F}_{2},{V}_{2}}\right) \) be a local parametri-sation of \( {S}_{2} \) around \( f\left( p\right) \) . After possibly red... | Yes |
Example 3.2.8 Let \( A : {\mathbb{R}}^{3} \rightarrow {\mathbb{R}}^{3} \) be an orthogonal map, i.e. \( A \in \mathrm{O}\left( 3\right) \) . We set \( f : {S}^{2} \rightarrow {S}^{2}, f \mathrel{\text{:=}} {\left. A\right| }_{{S}^{2}} \) . Let \( p \in {S}^{2} \) . We find the differential of \( f \) at \( p \) . | For this purpose let \( c : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow {S}^{2} \) be a smooth parametrised curve with \( c\left( 0\right) = p \) and \( \dot{c}\left( 0\right) = X \in {T}_{p}{S}^{2} \) . Because of the linearity of \( A \) we have\n\n\[ \n{\left. \frac{d}{dt}\left( f \circ c\right) \right| }... | Yes |
Example 3.3.1 Let \( S \subset {\mathbb{R}}^{3} \) be a plane. Then \( S \) can be described by an affine-linear parametrisation as in example 3.1.3, | \[ F : {\mathbb{R}}^{2} \rightarrow {\mathbb{R}}^{3} \] \[ F\left( {{u}^{1},{u}^{2}}\right) = {p}_{0} + {u}^{1} \cdot X + {u}^{2} \cdot Y,\;{p}_{0}, X, Y \in {\mathbb{R}}^{3}. \] Hence \( S \) is here the plane spanned by the vectors \( X \) and \( Y \) through the point \( {p}_{0} \). We find the first fundamental for... | Yes |
Example 3.3.2 Let us consider the cylindrical surface\n\n\[ S = \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3} \\mid {x}^{2} + {y}^{2} = 1}\\right\\} .\n\] | We use the local parametrisation\n\n\[ F : \\left( {0,{2\\pi }}\\right) \\times \\mathbb{R} \\rightarrow {\\mathbb{R}}^{3},\n\]\n\n\[ F\\left( {\\varphi, h}\\right) = \\left( \\begin{matrix} \\cos \\left( \\varphi \\right) \\\\ \\sin \\left( \\varphi \\right) \\\\ h \\end{matrix}\\right) .\n\]\n\nFor the first fundamen... | Yes |
Example 3.3.3 We find the first fundamental form of the sphere\n\n\[ \n{S}^{2} = \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3}\\left| {\\;{x}^{2} + {y}^{2} + {z}^{2} = 1}\\right. }\\right\\} \n\]\n\nin polar coordinates \( \\left( {{u}^{1},{u}^{2}}\\right) = \\left( {\\theta ,\\varphi }\\right) ... | From\n\n\[ \n\\frac{\\partial F}{\\partial \\theta }\\left( {\\theta ,\\varphi }\\right) = \\left( \\begin{matrix} - \\sin \\left( \\theta \\right) \\cdot \\cos \\left( \\varphi \\right) \\\\ - \\sin \\left( \\theta \\right) \\cdot \\sin \\left( \\varphi \\right) \\\\ \\cos \\left( \\theta \\right) \\end{matrix}\\right... | Yes |
Example 3.4.3 Let \( S = {S}^{2} \) . Then we obtain a unit normal field by setting | \( N = \mathrm{{Id}} \) | Yes |
Example 3.5.2 Let \( S = {S}^{2} \), and let \( N \) be the outer unit normal field, \( N\left( p\right) = p \) . Then | \[ {W}_{p} = - \operatorname{Id} : {T}_{p}{S}^{2} \rightarrow {T}_{p}{S}^{2} \] | No |
Example 3.5.4 Let \( S = {S}^{1} \times \mathbb{R} \) be the cylinder, \( N\left( {x, y, z}\right) = {\left( x, y,0\right) }^{\top } \). At a point \( p = {\left( x, y, z\right) }^{\top } \in S \) the tangent plane \( {T}_{p}S \) is spanned by the basis vectors \( {\left( -y, x,0\right) }^{\top } \) and \( {\left( 0,0,... | \n\[
{W}_{p}\left( \begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right) = - {d}_{p}N\left( \begin{array}{l} 0 \\ 0 \\ 1 \end{array}\right) = - {\left. \frac{d}{dt}N\left( \begin{matrix} x \\ y \\ z + t \end{matrix}\right) \right| }_{t = 0}
\]
\[
= - {\left. \frac{d}{dt}\left( \begin{array}{l} x \\ y \\ 0 \end{array}\right)... | Yes |
Theorem 3.6.1 (Meusnier’s theorem) Let \( S \subset {\mathbb{R}}^{3} \) be an orientable regular surface with unit normal field \( N \) and second fundamental form II. Let \( p \in S \). Let \( c : \left( {-\varepsilon ,\varepsilon }\right) \rightarrow S \) be a curve parametrised by arc-length with \( c\left( 0\right)... | Proof As \( c \) lies on \( S \), we have\n\n\[ \n\langle N\left( {c\left( t\right) }\right) ,\dot{c}\left( t\right) \rangle = 0 \n\]\n\nfor all \( t \in \left( {-\varepsilon ,\varepsilon }\right) \). Differentiating this equation gives\n\n\[ \n0 = {\left. \frac{d}{dt}\langle N\left( c\left( t\right) \right) ,\dot{c}\l... | Yes |
Let \( S = {S}^{1} \times \mathbb{R} \) be the cylinder. Let \( p \in S \). The intersection of \( S \) with the normal plane at the point \( p \) is a circle, an ellipse or a straight line. The normal curvature therefore varies between 1 and 0, depending on the direction. | By proposition 3.5.5 the Weingarten map \( {W}_{p} : {T}_{p}S \rightarrow {T}_{p}S \) is always selfadjoint. We can therefore find an orthonormal basis \( {X}_{1},{X}_{2} \) of \( {T}_{p}S \), which consists of eigenvectors of \( {W}_{p} \) only, \[ {W}_{p}\left( {X}_{i}\right) = {\kappa }_{i} \cdot {X}_{i},\;i = 1,2. ... | Yes |
Example 3.6.6 Let \( S = {S}^{1} \times \mathbb{R} \) be the cylinder, \( p = {\left( x, y, z\right) }^{\top } \). As we have seen, the Weingarten map \( {W}_{p} \) with respect to the inner unit normal field and the basis \( {X}_{1} = {\left( -y, x,0\right) }^{\top } \) and \( {X}_{2} = {\left( 0,0,1\right) }^{\top } ... | \[ \left( \begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right) \] This means precisely that \( {X}_{1} \) and \( {X}_{2} \) are principal curvature directions for the principal curvatures \( {\kappa }_{1} = 1 \) and \( {\kappa }_{2} = 0 \). | Yes |
The hyperbolic paraboloid \[ S = \left\{ {{\left( x, y, z\right) }^{\top } \in {\mathbb{R}}^{3}\left| {\;z = {y}^{2} - {x}^{2}}\right. }\right\} \] is the graph of the function \( \varphi : {\mathbb{R}}^{2} \rightarrow \mathbb{R},\varphi \left( {x, y}\right) = {y}^{2} - {x}^{2} \), and hence a regular surface. | To find a normal field, we write \( S \) as a set of zeros \( S = {f}^{-1}\left( 0\right) \), where \( f : {\mathbb{R}}^{3} \rightarrow \) \( \mathbb{R}, f\left( {x, y, z}\right) = z - {y}^{2} + {x}^{2} \) . The gradient is then \[ \operatorname{grad}f\left( {x, y, z}\right) = \left( \begin{array}{r} {2x} \\ - {2y} \\ ... | Yes |
Theorem 3.6.15 Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface, let \( p \in S \) and let \( {X}_{1},{X}_{2} \) be an orthonormal basis of \( {T}_{p}S \) . Let \( N \) be a smooth unit normal field on \( S \), defined on a neighbourhood of the point \( p \), such that \( \left( {{X}_{1},{X}_{2}, N\left( p\rig... | Proof Let us for now begin with an arbitrary local parametrisation \( \left( {{U}_{1},{F}_{1},{V}_{1}}\right) \) of \( S \) around \( p \) . We will repeatedly transform this parametrisation, and step by step establish the desired properties.\n\n(a) Let \( {x}_{0} \in {U}_{1} \) be the point for which \( {F}_{1}\left( ... | No |
Corollary 3.6.16 Every regular surface \( S \) can locally be given as a graph on its affine tangent plane \( {T}_{p}S + p \) . | Proof Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface, \( p \in S \) . For simplicity we rotate and translate the surface in \( {\mathbb{R}}^{3} \) in such a way that \( p = {\left( 0,0,0\right) }^{\top } \) and that the tangent plane \( {T}_{p}S \) is spanned precisely by the first two unit vectors \( {e}_{1... | Yes |
Lemma 3.7.2 Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface, let \( \left( {U, F, V}\right) \) and \( \left( {\widetilde{U},\widetilde{F},\widetilde{V}}\right) \) be local parametrisations of \( S \) . Let \( f : S \rightarrow \mathbb{R} \) be a function satisfying \( {\left. f\right| }_{S - \left( {V \cap \w... | Proof Let \( \varphi \mathrel{\text{:=}} {\widetilde{F}}^{-1} \circ F \) be the parameter transformation. By (3.2) we have\n\n\[ \left( {g}_{ij}\right) = {\left( D\varphi \right) }^{\top } \cdot \left( {{\widetilde{g}}_{ij} \circ \varphi }\right) \cdot {D\varphi }, \]\n\nhence\n\n\[ \det \left( {g}_{ij}\right) = \det \... | Yes |
If \( S = {\mathbb{R}}^{2} \times \{ 0\} \subset {\mathbb{R}}^{3} \) is the \( x - y \) plane, then we choose the Cartesian coordinates \( U = {\mathbb{R}}^{2}, V = {\mathbb{R}}^{3}, F\left( {x, y}\right) = {\left( x, y,0\right) }^{\top } \) . Then \( \left( {{g}_{ij}\left( {x, y}\right) }\right) = \) \( \left( \begin{... | If one wants to integrate the function in polar coordinates \( r \) and \( \varphi \), given by the local parametrisation\n\n\[ \n\widetilde{F} : \left( {0,\infty }\right) \times \left( {0,{2\pi }}\right) \rightarrow {\mathbb{R}}^{3} \n\]\n\n\[ \n\widetilde{F}\left( {r,\varphi }\right) = {\left( r \cdot \cos \varphi, r... | Yes |
Example 3.7.8 Let \( I \subset \mathbb{R} \) be an open interval, and \( c : I \rightarrow {\mathbb{R}}^{2} \) a plane parametrised regular curve. Let \( h > 0 \). We consider the generalised cylinder over \( c \): \[ S = \left\{ {c\left( t\right) + s{e}_{3} \mid t \in I,0 < s < h}\right\} . \] We can cover \( S \) wit... | For the differential of \( F \) we have \[ {D}_{\left( t, s\right) }F = \left( {\dot{c}\left( t\right) ,{e}_{3}}\right) . \] It follows that \[ {g}_{11}\left( {t, s}\right) = \langle \dot{c}\left( t\right) ,\dot{c}\left( t\right) \rangle \] \[ {g}_{12}\left( {t, s}\right) = {g}_{21}\left( {t, s}\right) = \left\langle {... | Yes |
Example 3.7.9 We calculate the area of the sphere \( S = {S}^{2} \) . We use polar coordinates | \[ U = \left( {0,{2\pi }}\right) \times \left( {\frac{\pi }{2},\frac{\pi }{2}}\right) ,\;V = {\mathbb{R}}^{3} - \left\{ {{\left( x, y, z\right) }^{\top } \mid x \geq 0, y = 0}\right\} , \] \[ F\left( {\varphi ,\vartheta }\right) = {\left( \cos \left( \varphi \right) \cos \left( \vartheta \right) ,\sin \left( \varphi \r... | Yes |
Example 3.8.2 If \( c : I \rightarrow {\mathbb{R}}^{3} \) is a plane parametrised curve that does not intersect itself, \( c\left( t\right) = \left( {{c}_{1}\left( t\right) ,{c}_{2}\left( t\right) ,0}\right) \) and \( v\left( t\right) = \left( {0,0,1}\right) \), then the corresponding ruled surface | \[ F\left( {t, s}\right) = \left( \begin{matrix} {c}_{1}\left( t\right) \\ {c}_{2}\left( t\right) \\ s \end{matrix}\right) \] is the generalised cylinder over \( c \) . We can take \( U = I \times \mathbb{R} \) as the domain. Compare example 3.7.8. | Yes |
Example 3.8.3 Let us consider a plane parametrised curve \( c : I \rightarrow {\mathbb{R}}^{3} \) that does not intersect itself, \( c\left( t\right) = \left( {{c}^{1}\left( t\right) ,{c}^{2}\left( t\right) ,0}\right) \) . For a fixed point \( p \in {\mathbb{R}}^{3} - \left( {{\mathbb{R}}^{2} \times }\right. \) \( \{ 0... | \[ F : I \times \left( {-\infty ,1}\right) \rightarrow {\mathbb{R}}^{3},\;F\left( {t, s}\right) = \left( {1 - s}\right) c\left( t\right) + {sp} \] is the generalised cone over \( c \) with apex \( p \) . | Yes |
Example 3.8.4 The Möbius strip is a ruled surface as well. We consider\n\n\[ F : \mathbb{R} \times \left( {-1,1}\right) \rightarrow {\mathbb{R}}^{3}, \]\n\n\[ F\left( {t, s}\right) = \left( \begin{matrix} \cos \left( t\right) + s \cdot \cos \left( t\right) \cos \left( {t/2}\right) \\ \sin \left( t\right) + s \cdot \sin... | The vector field \( v\left( t\right) = {\left( \cos \left( t\right) \cos \left( t/2\right) ,\sin \left( t\right) \cos \left( t/2\right) ,\sin \left( t/2\right) \right) }^{\top } \) rotates with half the speed from \( {\left( 1,0,0\right) }^{\top } \) to \( {\left( -1,0,0\right) }^{\top } \) while \( t \) runs through o... | Yes |
Example 3.8.5 It may be surprising that the hyperboloid of revolution, also called the one-sheeted hyperboloid or hyperboloid of one sheet,\n\n\[ S = \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3}\\left| {\\;1 + {z}^{2} = {x}^{2} + {y}^{2}}\\right. }\\right\\} ,\]\n\nis a ruled surface. | However, it is easy to check that the ruled surface given by\n\n\[ c\\left( t\\right) = {\\left( \\cos \\left( t\\right) ,\\sin \\left( t\\right) ,0\\right) }^{\\top },\]\n\n\[ v\\left( t\\right) = \\dot{c}\\left( t\\right) + {e}_{3} = {\\left( -\\sin \\left( t\\right) ,\\cos \\left( t\\right) ,1\\right) }^{\\top },\]\... | Yes |
Example 3.8.6 The hyperbolic paraboloid or saddle surface\n\n\[ S = \\left\\{ {{\\left( x, y, z\\right) }^{\\top } \\in {\\mathbb{R}}^{3} \\mid z = {xy}} \\right\\} \] is a ruled surface as well. | This is the case since\n\n\[ c\\left( t\\right) = {\\left( t,0,0\\right) }^{\\top } \]\n\n\[ v\\left( t\\right) = \\frac{1}{\\sqrt{1 + {t}^{2}}}{\\left( 0,1, t\\right) }^{\\top }.\] | Yes |
Theorem 3.8.7 Let \( S \subset {\mathbb{R}}^{3} \) be a ruled surface. Then the Gauss curvature\n\nsatisfies\n\n\[ K \leq 0. \] | Proof Let us remember that the Gauss curvature can be given in terms of the determinants of the first and second fundamental forms,\n\n\[ K = \frac{\det \left( {h}_{ij}\right) }{\det \left( {g}_{ij}\right) } \]\n\nAs the first fundamental form is \( \left( {g}_{ij}\right) \) positive definite, we have in particular tha... | Yes |
Corollary 3.8.9 Let \( S \subset {\mathbb{R}}^{3} \) be a regular surface with compact closure \( \bar{S} \) . We assume that \( S \) has minimal area among all regular surfaces \( \widetilde{S} \) with the same boundary \( \partial \widetilde{S} = \partial S \) . Then the mean curvature field of \( S \) satisfies\n\n\... | Proof Suppose that \( \mathcal{H}\left( p\right) \neq {\left( 0,0,0\right) }^{\top } \) for a point \( p \in S \) . In a neighbourhood of \( p \) we consider the smooth unit normal field \( N \), for which \( \langle \mathcal{H}\left( p\right), N\left( p\right) \rangle > 0 \) . For reasons of continuity \( \langle \mat... | Yes |
Example 3.8.11 The simplest and most uninteresting example is certainly the affine plane \( S \subset {\mathbb{R}}^{3} \) . For this surface we obviously have | \[ K \equiv H \equiv 0. \] | Yes |
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