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Lemma 6.4. If \( \gamma \) is an admissible curve and \( V \) is a vector field along \( \gamma \) , then \( V \) is the variation field of some variation of \( \gamma \) . If \( V \) is proper, the variation can be taken to be proper as well. | Proof. Set \( \Gamma \left( {s, t}\right) = \exp \left( {{sV}\left( t\right) }\right) \) (Figure 6.5). By compactness of \( \left\lbrack {a, b}\right\rbrack \), there is some positive \( \varepsilon \) such that \( \Gamma \) is defined on \( \left( {-\varepsilon ,\varepsilon }\right) \times \left\lbrack {a, b}\right\rb... | Yes |
Theorem 6.6. Every minimizing curve is a geodesic when it is given a unit speed parametrization. | Proof. Suppose \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow M \) is minimizing and unit speed, and let \( a = \) \( {a}_{0} < \cdots < {a}_{k} = b \) be a subdivision such that \( \gamma \) is smooth on \( \left\lbrack {{a}_{i - 1},{a}_{i}}\right\rbrack \) . If \( \Gamma \) is any proper variation of \( \ga... | Yes |
Corollary 6.7. A unit speed admissible curve \( \gamma \) is a critical point for \( L \) if and only if it is a geodesic. | Proof. If \( \gamma \) is a critical point, the proof of Theorem 6.6 goes through without modification to show that \( \gamma \) is a geodesic. Conversely, if \( \gamma \) is a geodesic, then the first term in the second variation formula vanishes by the geodesic equation, and the second term vanishes because \( \dot{\... | Yes |
Corollary 6.9. Let \( \left( {x}^{i}\right) \) be normal coordinates on a geodesic ball \( \mathcal{U} \) centered at \( p \in M \), and let \( r \) be the radial distance function as defined in (5.9). Then \( \operatorname{grad}r = \partial /\partial r \) on \( \mathcal{U} - \{ p\} \) . | Proof. For any \( q \in \mathcal{U} - \{ p\} \) and \( Y \in {T}_{q}M \), we need to show that\n\n\[ \n{dr}\left( Y\right) = \left\langle {\frac{\partial }{\partial r}, Y}\right\rangle \n\]\n\n(6.4)\n\nThe geodesic sphere \( {\exp }_{p}\left( {\partial {B}_{R}\left( 0\right) }\right) \) through \( q \) is characterized... | Yes |
Corollary 6.11. Within any geodesic ball around \( p \in M \), the radial distance function \( r\left( x\right) \) defined by (5.9) is equal to the Riemannian distance from \( p \) to \( x \) . | Proof. The radial geodesic \( \gamma \) from \( p \) to \( x \) is minimizing by Proposition 6.10. Since its velocity is equal to \( \partial /\partial r \), which is a unit vector in both the \( g \) norm and the Euclidean norm in normal coordinates, the \( g \) -length of \( \gamma \) is equal to its Euclidean length... | Yes |
Theorem 6.12. Every Riemannian geodesic is locally minimizing. | Proof. Let \( \gamma : I \rightarrow M \) be a geodesic, which we may assume to be defined on an open interval, and let \( {t}_{0} \in I \) . Let \( \mathcal{W} \) be a uniformly normal neighborhood of \( \gamma \left( {t}_{0}\right) \), and let \( \mathcal{U} \subset I \) be the connected component of \( {\gamma }^{-1... | Yes |
Theorem 6.13. (Hopf-Rinow) A connected Riemannian manifold is geodesically complete if and only if it is complete as a metric space. | Proof. Suppose first that \( M \) is complete as a metric space but not geodesically complete. Then there is some unit speed geodesic \( \gamma : \lbrack 0, b) \rightarrow M \) that extends to no interval \( \lbrack 0, b + \varepsilon ) \) for \( \varepsilon > 0 \) . Let \( \left\{ {t}_{i}\right\} \) be any increasing ... | Yes |
Corollary 6.14. If there exists one point \( p \in M \) such that the restricted exponential map \( {\exp }_{p} \) is defined on all of \( {T}_{p}M \), then \( M \) is complete. | Null | No |
Corollary 6.15. \( M \) is complete if and only if any two points in \( M \) can be joined by a minimizing geodesic segment. | Null | No |
Corollary 6.16. If \( M \) is compact, then every geodesic can be defined for all time. | Null | No |
Lemma 7.2. The Riemann curvature endomorphism and curvature tensor are local isometry invariants. More precisely, if \( \varphi : \left( {M, g}\right) \rightarrow \left( {\widetilde{M},\widetilde{g}}\right) \) is a local isometry, then\n\n\[{\varphi }^{ * }\widetilde{Rm} = {Rm}\]\n\n\[ \widetilde{R}\left( {{\varphi }_{... | Exercise 7.2. Prove Lemma 7.2. | No |
Proposition 7.4. (Symmetries of the Curvature Tensor) The curvature tensor has the following symmetries for any vector fields \( W, X, Y \) , \( Z \) :\n\n(a) \( \operatorname{Rm}\left( {W, X, Y, Z}\right) = - \operatorname{Rm}\left( {X, W, Y, Z}\right) \) .\n\n(b) \( \operatorname{Rm}\left( {W, X, Y, Z}\right) = - \op... | Proof of Proposition 7.4. Identity (a) is immediate from the obvious fact that \( R\left( {W, X}\right) Y = - R\left( {X, W}\right) Y \) . To prove (b), it suffices to show that \( {Rm}\left( {W, X, Y, Y}\right) = 0 \) for all \( Y \), for then (b) follows from the expansion of \( \operatorname{Rm}\left( {W, X, Y + Z, ... | Yes |
Proposition 7.5. (Differential Bianchi Identity) The total covariant derivative of the curvature tensor satisfies the following identity:\n\n\[ \n\nabla {Rm}\left( {X, Y, Z, V, W}\right) + \nabla {Rm}\left( {X, Y, V, W, Z}\right) + \nabla {Rm}\left( {X, Y, W, Z, V}\right) = 0.\n\] | Proof. First of all, by the symmetries of \( {Rm} \) ,(7.6) is equivalent to\n\n\[ \n\nabla {Rm}\left( {Z, V, X, Y, W}\right) + \nabla {Rm}\left( {V, W, X, Y, Z}\right) + \nabla {Rm}\left( {W, Z, X, Y, V}\right) = 0.\n\]\n\nThis can be proved by a long and tedious computation, but there is a standard shortcut for such ... | Yes |
Exercise 7.5. Prove Lemma 7.6, using the symmetries of the curvature tensor. | Null | No |
Lemma 7.7. (Contracted Bianchi Identity) The covariant derivatives of the Ricci and scalar curvatures satisfy the following identity:\n\n\[ \operatorname{div}{Rc} = \frac{1}{2}\nabla S \]\n\nwhere div is the divergence operator (Problem 3-3). In components, this is\n\n\[ {R}_{{ij};}{}^{j} = \frac{1}{2}{S}_{;i} \] | Proof. Formula (7.9) follows immediately by contracting the component form (7.7) of the differential Bianchi identity on the indices \( i, l \) and then again on \( j, k \), after raising one index of each pair. | No |
Proposition 7.8. If \( g \) is an Einstein metric on a connected manifold of dimension \( n \geq 3 \), its scalar curvature is constant. | Proof. Taking the covariant derivative of each side of (7.10) and noting that the covariant derivative of the metric is zero, we see that the Einstein condition implies\n\n\[ {R}_{{ij};k} = \frac{1}{n}{S}_{;k}{g}_{ij} \]\n\nTracing this equation on \( j \) and \( k \), and comparing with the contracted Bianchi identity... | Yes |
Lemma 8.1. The second fundamental form is\n\n(a) independent of the extensions of \( X \) and \( Y \) ;\n\n(b) bilinear over \( {C}^{\infty }\left( M\right) \) ; and\n\n(c) symmetric in \( X \) and \( Y \) . | Proof. First we show that the symmetry of \( \Pi \) follows from the symmetry of the connection \( \widetilde{\nabla } \) . Let \( X \) and \( Y \) be extended arbitrarily to \( M \) . Then\n\n\[ \Pi \left( {X, Y}\right) - \Pi \left( {Y, X}\right) = {\left( {\widetilde{\nabla }}_{X}Y - {\widetilde{\nabla }}_{Y}X\right)... | Yes |
Theorem 8.2. (The Gauss Formula) If \( X, Y \in \mathcal{T}\left( M\right) \) are extended arbitrarily to vector fields on \( \widetilde{M} \), the following formula holds along \( M \) : | Proof. Because of the decomposition (8.1) and the definition of the second fundamental form, it suffices to show that \( {\left( {\widetilde{\nabla }}_{X}Y\right) }^{\top } = {\nabla }_{X}Y \) at all points of \( M \) .\n\nDefine a map \( {\nabla }^{\top } : \mathcal{T}\left( M\right) \times \mathcal{T}\left( M\right) ... | Yes |
Lemma 8.3. (The Weingarten Equation) Suppose \( X, Y \in \mathfrak{T}\left( M\right) \) and \( N \in \mathcal{N}\left( M\right) \) . When \( X, Y, N \) are extended arbitrarily to \( \widetilde{M} \), the following equation holds at points of \( M \) : | Proof. Since \( \langle N, Y\rangle \) vanishes identically along \( M \) and \( X \) is tangent to \( M \) , the following holds along \( M \) :\n\n\[ 0 = X\langle N, Y\rangle \]\n\n\[ = \left\langle {{\widetilde{\nabla }}_{X}N, Y}\right\rangle + \left\langle {N,{\widetilde{\nabla }}_{X}Y}\right\rangle \]\n\n\[ = \lef... | Yes |
Theorem 8.4. (The Gauss Equation) For any \( X, Y, Z, W \in {T}_{p}M \), the following equation holds:\n\n\[ \widetilde{\operatorname{Rm}}\left( {X, Y, Z, W}\right) = \operatorname{Rm}\left( {X, Y, Z, W}\right) \]\n\n\[ - \langle \Pi \left( {X, W}\right) ,\Pi \left( {Y, Z}\right) \rangle + \langle \Pi \left( {X, Z}\rig... | Proof. Let \( X, Y, Z, W \) be extended arbitrarily to vector fields on \( M \), and then to vector fields on \( \widetilde{M} \) that are tangent to \( M \) at points of \( M \) . Along \( M \), the Gauss formula gives\n\n\[ \widetilde{Rm}\left( {X, Y, Z, W}\right) = \left\langle {{\widetilde{\nabla }}_{X}{\widetilde{... | Yes |
Lemma 8.5. (The Gauss Formula Along a Curve) Let \( M \) be a Riemannian submanifold of \( \widetilde{M} \), and \( \gamma \) a curve in \( M \) . For any vector field \( V \) tangent to \( M \) along \( \gamma \) ,\n\n\[{\widetilde{D}}_{t}V = {D}_{t}V + \Pi \left( {\dot{\gamma }, V}\right)\] | Proof. In terms of an adapted orthonormal frame, \( V \) can be written \( V\left( t\right) = \) \( {V}^{i}\left( t\right) {E}_{i} \), where the sum is only over \( i = 1,\ldots, n \) . Applying the product rule and the Gauss formula, we get\n\n\[{\widetilde{D}}_{t}V = {\dot{V}}^{i}{E}_{i} + {V}^{i}{\widetilde{\nabla }... | Yes |
Theorem 8.6. (Gauss’s Theorema Egregium) Let \( M \subset {\mathbf{R}}^{3} \) be a 2-dimensional submanifold and \( g \) the induced metric on \( M \). For any \( p \in M \) and any basis \( \left( {X, Y}\right) \) for \( {T}_{p}M \), the Gaussian curvature of \( M \) at \( p \) is given \( {by} \)\n\n\[ K = \frac{\ope... | Proof. We begin with the special case in which \( \left( {X, Y}\right) = \left( {{E}_{1},{E}_{2}}\right) \) is an orthonormal basis for \( {T}_{p}M \). In this case the denominator in (8.5) is equal to 1. If we write \( {h}_{ij} = \dot{h}\left( {{E}_{i},{E}_{j}}\right) \), then in this basis \( K = \det s = \det \left(... | Yes |
Lemma 8.7. The Gaussian curvature of a Riemannian 2-manifold is related to the curvature tensor, Ricci tensor, and scalar curvature by the formulas\n\n\[ \n{Rm}\left( {X, Y, Z, W}\right) = K\left( {\langle X, W\rangle \langle Y, Z\rangle -\langle X, Z\rangle \langle Y, W\rangle }\right) \]\n\n\[ \n{Rc}\left( {X, Y}\rig... | Proof. Since both sides of the first equation are tensors, we can compute them in terms of any basis. Let \( \left( {{E}_{1},{E}_{2}}\right) \) be any orthonormal basis for \( {T}_{p}M \) , and consider the components \( {R}_{ijkl} = {Rm}\left( {{E}_{i},{E}_{j},{E}_{k},{E}_{l}}\right) \) of the curvature tensor. In ter... | Yes |
Proposition 8.8. If \( \\left( {X, Y}\\right) \) is any basis for a 2-plane \( \\Pi \\subset {T}_{p}M \), then\n\n\[ K\\left( {X, Y}\\right) = \\frac{\\operatorname{Rm}\\left( {X, Y, Y, X}\\right) }{{\\left| X\\right| }^{2}{\\left| Y\\right| }^{2}-\\langle X, Y{\\rangle }^{2}}.\] | Proof. For this proof, we denote the induced metric on \( {S}_{\\Pi } \) by \( \\widetilde{g} \), and continue to denote the metric on \( M \) by \( g \) . As in the first part of this chapter, we use tildes to denote geometric quantities associated with \( \\widetilde{g} \), but note that now the roles of \( g \) and ... | Yes |
Lemma 8.9. Suppose \( {\mathcal{R}}_{1} \) and \( {\mathcal{R}}_{2} \) are covariant 4-tensors on a vector space \( V \) with an inner product, and both have the symmetries of the curvature tensor (as described in Proposition 7.4). If for every pair of independent vectors \( X, Y \in V \), \[ \frac{{\mathcal{R}}_{1}\le... | Proof. Setting \( \mathcal{R} = {\mathcal{R}}_{1} - {\mathcal{R}}_{2} \), it suffices to show \( \mathcal{R} = 0 \) under the assumption that \( \mathcal{R}\left( {X, Y, Y, X}\right) = 0 \) for all \( X, Y \). For any vectors \( X, Y, Z \), since \( \mathcal{R} \) also has the symmetries of the curvature tensor, \[ 0 =... | Yes |
Lemma 8.10. Suppose \( \left( {M, g}\right) \) is any Riemannian \( n \) -manifold with constant sectional curvature \( C \) . The curvature endomorphism, curvature tensor, Ricci tensor, and scalar curvature of \( g \) are given by the formulas\n\n\[ R\left( {X, Y}\right) Z = C\left( {\\langle Y, Z\\rangle X-\\langle X... | Exercise 8.8. Prove Lemma 8.10. | No |
Lemma 9.2. If \( \gamma \) is a positively oriented curved polygon in \( M \), the rotation angle of \( \gamma \) is \( {2\pi } \) . | Proof. If we use the given coordinate chart to consider \( \gamma \) as a curved polygon in the plane, we can compute its tangent angle function either with respect to \( g \) or with respect to the Euclidean metric \( \bar{g} \) . In either case, \( \operatorname{Rot}\left( \gamma \right) \) is an integral multiple of... | Yes |
Corollary 9.4. (Angle-Sum Theorem) The sum of the interior angles of a Euclidean triangle is \( \pi \) . | Null | No |
Corollary 9.5. (Circumference Theorem) The circumference of a Euclidean circle of radius \( R \) is \( {2\pi R} \) . | Null | No |
Corollary 9.6. (Total Curvature Theorem) If \( \gamma : \left\lbrack {a, b}\right\rbrack \rightarrow {\mathbf{R}}^{2} \) is a unit speed simple closed curve such that \( \dot{\gamma }\left( a\right) = \dot{\gamma }\left( b\right) \), and \( N \) is the inward-pointing normal, then\n\n\[{\int }_{a}^{b}{\kappa }_{N}\left... | Null | No |
Theorem 9.7. (The Gauss-Bonnet Theorem) If \( M \) is a triangulated, compact, oriented, Riemannian 2-manifold, then\n\n\[{\int }_{M}{KdA} = {2\pi \chi }\left( M\right)\] | Proof. Let \( \left\{ {{\Omega }_{i} : i = 1,\ldots ,{N}_{f}}\right\} \) denote the faces of the triangulation, and for each \( i \) let \( \left\{ {{\gamma }_{ij} : j = 1,2,3}\right\} \) be the edges of \( {\Omega }_{i} \) and \( \left\{ {{\theta }_{ij} : j = 1,2,3}\right\} \) its interior angles. Since each exterior ... | Yes |
Corollary 9.8. Let \( M \) be a compact Riemannian 2-manifold and \( K \) its Gaussian curvature.\n\n(a) If \( M \) is homeomorphic to the sphere or the projective plane, then \( K > 0 \) somewhere.\n\n(b) If \( M \) is homeomorphic to the torus or the Klein bottle, then either \( K \equiv 0 \) or \( K \) takes on both... | Proof. If \( M \) is orientable, the result follows immediately from the Gauss-Bonnet theorem, because a function whose integral is positive, negative, or zero must satisfy the claimed sign condition. If \( M \) is nonorientable, the result follows by applying the Gauss-Bonnet theorem to the orientable double cover \( ... | Yes |
Exercise 9.2. Prove Corollary 9.9. | Null | No |
Lemma 10.1. If \( \\Gamma \) is any smooth admissible family of curves, and \( V \) is a smooth vector field along \( \\Gamma \), then\n\n\[ \n{D}_{s}{D}_{t}V - {D}_{t}{D}_{s}V = R\\left( {S, T}\\right) V.\n\] | Proof. This is a local issue, so we can compute in any local coordinates.\n\nWriting \( V\\left( {s, t}\\right) = {V}^{i}\\left( {s, t}\\right) {\\partial }_{i} \), we compute\n\n\[ \n{D}_{t}V = \\frac{\\partial {V}^{i}}{\\partial t}{\\partial }_{i} + {V}^{i}{D}_{t}{\\partial }_{i}\n\]\n\nTherefore,\n\n\[ \n{D}_{s}{D}_... | Yes |
Theorem 10.2. (The Jacobi Equation) Let \( \gamma \) be a geodesic and \( V \) a vector field along \( \gamma \) . If \( V \) is the variation field of a variation through geodesics, then \( V \) satisfies\n\n\[ \n{D}_{t}^{2}V + R\left( {V,\dot{\gamma }}\right) \dot{\gamma } = 0 \n\] | Proof. With \( S \) and \( T \) as before, the preceding lemma implies\n\n\[ \n0 = {D}_{s}{D}_{t}T \n\]\n\n\[ \n= {D}_{t}{D}_{s}T + R\left( {S, T}\right) T \n\]\n\n\[ \n= {D}_{t}{D}_{t}S + R\left( {S, T}\right) T \n\]\n\nwhere the last step follows from the symmetry lemma. Evaluating at \( s = 0 \) , where \( S\left( {... | Yes |
Lemma 10.3. Every Jacobi field along a geodesic \( \gamma \) is the variation field of some variation of \( \gamma \) through geodesics. | Exercise 10.1. Prove Lemma 10.3. [Hint: Let \( \Gamma \left( {s, t}\right) = {\exp }_{\sigma \left( s\right) }{tW}\left( s\right) \) for a suitable curve \( \sigma \) and vector field \( W \) along \( \sigma \) .] | No |
Proposition 10.4. (Existence and Uniqueness of Jacobi Fields) Let \( \gamma : I \rightarrow M \) be a geodesic, \( a \in I \), and \( p = \gamma \left( a\right) \) . For any pair of vectors \( X, Y \in {T}_{p}M \), there is a unique Jacobi field \( J \) along \( \gamma \) satisfying the initial conditions\n\n\[ J\left(... | Proof. Choose an orthonormal basis \( \left\{ {E}_{i}\right\} \) for \( {T}_{p}M \), and extend it to a parallel orthonormal frame along all of \( \gamma \) . Writing \( J\left( t\right) = {J}^{i}\left( t\right) {E}_{i} \), we can express the Jacobi equation as\n\n\[ {\ddot{J}}^{i} + {R}_{jkl}{}^{i}{J}^{j}{\dot{\gamma ... | Yes |
Corollary 10.5. Along any geodesic \( \gamma \), the set of Jacobi fields is a \( {2n} \) - dimensional linear subspace of \( \mathcal{T}\left( \gamma \right) \) . | Proof. Let \( p = \gamma \left( a\right) \) be any point on \( \gamma \), and consider the map from the set of Jacobi fields along \( \gamma \) to \( {T}_{p}M \oplus {T}_{p}M \) by sending \( J \) to \( \left( {J\left( a\right) ,{D}_{t}J\left( a\right) }\right) \) . The preceding proposition says precisely that this ma... | Yes |
Lemma 10.6. Let \( \gamma : I \rightarrow M \) be a geodesic, and \( a \in I \) .\n\n(a) A Jacobi field J along \( \gamma \) is normal if and only if\n\n\[ J\left( a\right) \bot \dot{\gamma }\left( a\right) \text{ and }{D}_{t}J\left( a\right) \bot \dot{\gamma }\left( a\right) . \] | Proof. Using compatibility with the metric and the fact that \( {D}_{t}\dot{\gamma } \equiv 0 \), we compute\n\n\[ \frac{{d}^{2}}{d{t}^{2}}\langle J,\dot{\gamma }\rangle = \left\langle {{D}_{t}^{2}J,\dot{\gamma }}\right\rangle \]\n\n\[ = - \langle R\left( {J,\dot{\gamma }}\right) \dot{\gamma },\dot{\gamma }\rangle \]\n... | Yes |
Lemma 10.7. Let \( p \in M \), let \( \left( {x}^{i}\right) \) be normal coordinates on a neighborhood \( \mathcal{U} \) of \( p \), and let \( \gamma \) be a radial geodesic starting at \( p \) . For any \( W = {W}^{i}{\partial }_{i} \in {T}_{p}M \), the Jacobi field \( J \) along \( \gamma \) such that \( J\left( 0\r... | Proof. An easy computation using formula (4.10) for covariant derivatives in coordinates shows that \( J \) satisfies the specified initial conditions, so it suffices to show that \( J \) is a Jacobi field. If we set \( V = \dot{\gamma }\left( 0\right) \in {T}_{p}M \), then we know from Lemma 5.11 that \( \gamma \) is ... | Yes |
Lemma 10.8. Suppose \( \\left( {M, g}\\right) \) is a Riemannian manifold with constant sectional curvature \( C \), and \( \\gamma \) is a unit speed geodesic in \( M \). The normal Jacobi fields along \( \\gamma \) vanishing at \( t = 0 \) are precisely the vector fields\n\n\[ J\\left( t\\right) = u\\left( t\\right) ... | Proof. Since \( g \) has constant curvature, its curvature endomorphism is given by the formula of Lemma 8.10:\n\n\[ R\\left( {X, Y}\\right) Z = C\\left( {\\langle Y, Z\\rangle X-\\langle X, Z\\rangle Y}\\right) \]\n\nSubstituting this into the Jacobi equation, we find that a normal Jacobi field \( J \) satisfies\n\n\[... | Yes |
Proposition 10.9. Suppose \( \left( {M, g}\right) \) is a Riemannian manifold with constant sectional curvature \( C \) . Let \( \left( {x}^{i}\right) \) be Riemannian normal coordinates on a normal neighborhood \( \mathcal{U} \) of \( p \in M \), let \( {\left| \cdot \right| }_{\bar{q}} \) be the Euclidean norm in the... | Proof. By the Gauss lemma, the decomposition \( V = {V}^{\top } + {V}^{ \bot } \) is orthogonal, so \( {\left| V\right| }_{g}^{2} = {\left| {V}^{ \bot }\right| }_{g}^{2} + {\left| {V}^{\top }\right| }_{g}^{2} \) . Since \( \partial /\partial r \) is a unit vector in both the \( g \) and \( \bar{g} \) norms, it is immed... | Yes |
Proposition 10.10. (Local Uniqueness of Constant Curvature Metrics) Let \( \left( {M, g}\right) \) and \( \left( {\widetilde{M},\widetilde{g}}\right) \) be Riemannian manifolds with constant sectional curvature \( C \) . For any points \( p \in M,\widetilde{p} \in \widetilde{M} \), there exist neighborhoods \( \mathcal... | Proof. Choose \( p \in M \) and \( \widetilde{p} \in \widetilde{M} \), and let \( \mathcal{U} \) and \( \widetilde{\mathcal{U}} \) be geodesic balls of small radius \( \varepsilon \) around \( p \) and \( \widetilde{p} \), respectively. Riemannian normal coordinates give maps \( \varphi : \mathcal{U} \rightarrow {B}_{\... | Yes |
Proposition 10.11. Suppose \( p \in M, V \in {T}_{p}M \), and \( q = {\exp }_{p}V \) . Then \( {\exp }_{p} \) is a local diffeomorphism in a neighborhood of \( V \) if and only if \( q \) is not conjugate to \( p \) along the geodesic \( \gamma \left( t\right) = {\exp }_{p}{tV},\;t \in \left\lbrack {0,1}\right\rbrack \... | Proof. By the inverse function theorem, \( {\exp }_{p} \) is a local diffeomorphism near \( V \) if and only if \( {\left( {\exp }_{p}\right) }_{ * } \) is an isomorphism at \( V \), and by dimensional considerations, this occurs if and only if \( {\left( {\exp }_{p}\right) }_{ * } \) is injective at \( V \) .\n\nIdent... | Yes |
Corollary 10.13. If \( \Gamma \) is a proper variation of a unit speed geodesic \( \gamma \) whose variation field is a proper normal vector field \( V \), the second variation of \( L\left( {\Gamma }_{s}\right) \) is \( I\left( {V, V}\right) \) . In particular, if \( \gamma \) is minimizing, then \( I\left( {V, V}\rig... | Null | No |
Proposition 10.14. For any pair of proper normal vector fields \( V, W \) along a geodesic segment \( \gamma \) , | Proof. On any subinterval \( \left\lbrack {{a}_{i - 1},{a}_{i}}\right\rbrack \) where \( V \) and \( W \) are smooth,\n\n\[ \frac{d}{dt}\left\langle {{D}_{t}V, W}\right\rangle = \left\langle {{D}_{t}^{2}V, W}\right\rangle + \left\langle {{D}_{t}V,{D}_{t}W}\right\rangle . \]\n\nThus, by the fundamental theorem of calcul... | Yes |
If \( \gamma \) is a geodesic segment from \( p \) to \( q \) that has an interior conjugate point to \( p \), then there exists a proper normal vector field \( X \) along \( \gamma \) such that \( I\left( {X, X}\right) < 0 \) . In particular, \( \gamma \) is not minimizing. | Proof. Suppose \( \gamma : \left\lbrack {0, b}\right\rbrack \rightarrow M \) is a unit speed parametrization of \( \gamma \), and \( \gamma \left( a\right) \) is conjugate to \( \gamma \left( 0\right) \) for some \( 0 < a < b \) . This means there is a nontrivial normal Jacobi field \( J \) along \( {\left. \gamma \rig... | Yes |
Theorem 11.1. (Sturm Comparison Theorem) Suppose \( u \) and \( v \) are differentiable real-valued functions on \( \left\lbrack {0, T}\right\rbrack \), twice differentiable on \( \left( {0, T}\right) \) , and \( u > 0 \) on \( \left( {0, T}\right) \) . Suppose further that \( u \) and \( v \) satisfy\n\n\[ \ddot{u}\le... | Proof. Consider the function \( f\left( t\right) = v\left( t\right) /u\left( t\right) \) defined on \( \left( {0, T}\right) \) . It follows from l’Hôpital’s rule that \( \mathop{\lim }\limits_{{t \rightarrow 0}}f\left( t\right) = \dot{v}\left( 0\right) /\dot{u}\left( 0\right) = 1 \) . Since \( f \) is differentiable on... | Yes |
Corollary 11.3. (Conjugate Point Comparison Theorem) Suppose all sectional curvatures of \( \left( {M, g}\right) \) are bounded above by a constant \( C \) . If\n\n\( C \leq 0 \), then no point of \( M \) has conjugate points along any geodesic. If \( C = 1/{R}^{2} > 0 \), then the first conjugate point along any geode... | Proof. If \( C \leq 0 \), the Jacobi field comparison theorem implies that any nontrivial normal Jacobi field vanishing at \( t = 0 \) satisfies \( \left| {J\left( t\right) }\right| > 0 \) for all \( t > 0 \) . Similarly, if \( C > 0 \), then \( \left| {J\left( t\right) }\right| \geq \left( \text{constant}\right) \sin ... | Yes |
Corollary 11.4. (Metric Comparison Theorem) Suppose all sectional curvatures of \( \left( {M, g}\right) \) are bounded above by a constant \( C \) . In any normal coordinate chart, \( g\left( {V, V}\right) \geq {g}_{C}\left( {V, V}\right) \), where \( {g}_{C} \) is the constant curvature metric given by formula (10.8). | Proof. Decomposing a vector \( V \) into components \( {V}^{\top } \) tangent to the geodesic sphere and \( {V}^{ \bot } \) tangent to the radial geodesics as in the proof of Proposition 10.9 gives\n\n\[ g\left( {V, V}\right) = g\left( {{V}^{ \bot },{V}^{ \bot }}\right) + g\left( {{V}^{\top },{V}^{\top }}\right) .\n\]\... | Yes |
Theorem 11.5. (The Cartan-Hadamard Theorem) If \( M \) is a complete, connected manifold all of whose sectional curvatures are nonpositive, then for any point \( p \in M,{\exp }_{p} : {T}_{p}M \rightarrow M \) is a covering map. In particular, the universal covering space of \( M \) is diffeomorphic to \( {\mathbf{R}}^... | Proof. The assumption of nonpositive curvature guarantees that \( p \) has no conjugate points along any geodesic, which can be shown by using either the conjugate point comparison theorem above or Problem 10-2. Therefore, by Proposition 10.11, \( {\exp }_{p} \) is a local diffeomorphism on all of \( {T}_{p}M \) . Let ... | Yes |
Lemma 11.6. Suppose \( \widetilde{M} \) and \( M \) are connected Riemannian manifolds, with \( \widetilde{M} \) complete, and \( \pi : \widetilde{M} \rightarrow M \) is a local isometry. Then \( M \) is complete and \( \pi \) is a covering map. | Proof. A fundamental property of covering maps is the path-lifting property: any continuous path \( \gamma \) in \( M \) lifts to a path \( \widetilde{\gamma } \) in \( \widetilde{M} \) such that \( \pi \circ \widetilde{\gamma } = \gamma \) . We begin by proving that \( \pi \) possesses the path-lifting property for ge... | Yes |
Theorem 11.8. (Myers’s Theorem) Suppose \( M \) is a complete, connected Riemannian n-manifold whose Ricci tensor satisfies the following inequality for all \( V \in {TM} \) :\n\n\[ \operatorname{Rc}\left( {V, V}\right) \geq \frac{n - 1}{{R}^{2}}{\left| V\right| }^{2} \]\n\nThen \( M \) is compact, with a finite fundam... | Proof. As in the proof of Bonnet's theorem, it suffices to prove the diameter estimate. As before, let \( \gamma \) be a minimizing unit speed geodesic segment of\n\nlength \( L > {\pi R} \) . Let \( \left( {{E}_{1},\ldots ,{E}_{n}}\right) \) be a parallel orthonormal frame along \( \gamma \) such that \( {E}_{n} = \do... | Yes |
Theorem 11.9. (Rauch Comparison Theorem) Let \( M \) and \( \widetilde{M} \) be Riemannian manifolds, let \( \gamma : \left\lbrack {0, T}\right\rbrack \rightarrow M \) and \( \widetilde{\gamma } : \left\lbrack {0, T}\right\rbrack \rightarrow \widetilde{M} \) be unit speed geodesic segments such that \( \widetilde{\gamm... | You can find proofs in [dC92], [CE75], and [Spi79, volume 4]. Letting \( \widetilde{M} \) be one of our constant curvature model spaces, we recover the Jacobi field comparison theorem above. On the other hand, if instead we take \( M \) to have constant curvature, we get the same result with the inequalities reversed. | No |
Theorem 11.10. (The Sphere Theorem) Suppose \( M \) is a complete, simply-connected, Riemannian n-manifold that is strictly \( \frac{1}{4} \) -pinched. Then \( M \) is homeomorphic to \( {\mathbf{S}}^{n} \) . | The proof, which can be found in [CE75] or [dC92], is an elaborate application of the Rauch comparison theorem together with the Morse index theorem mentioned in Chapter 10. This result is sharp, at least in even dimensions, because the Fubini-Study metrics on complex projective spaces are \( \frac{1}{4} \) -pinched (P... | Yes |
Theorem 11.11. (Hamilton) Suppose \( M \) is a simply-connected compact Riemannian 3-manifold with strictly positive Ricci curvature. Then \( M \) is diffeomorphic to \( {\mathbf{S}}^{3} \) . | Null | No |
Corollary 11.13. (Classification of Constant Curvature Metrics) Suppose \( M \) is a complete, connected Riemannian manifold with constant sectional curvature. Then \( M \) is isometric to \( \widetilde{M}/\Gamma \), where \( \widetilde{M} \) is one of the constant curvature model spaces \( {\mathbf{R}}^{n},{\mathbf{S}... | Proof. If \( \pi : \widetilde{M} \rightarrow M \) is the universal covering space of \( M \) with the lifted metric \( \widetilde{g} = {\pi }^{ * }g \), the preceding theorem shows that \( \left( {\widetilde{M},\widetilde{g}}\right) \) is isometric to one of the model spaces. From covering space theory [Sie92, Mas67] i... | Yes |
Lemma 1. Let \( S \) have density \( \alpha \) and \( 0 \in S \) . Then \( S \oplus S \) has density at least \( {2\alpha } - {\alpha }^{2} \) . | Proof. All the gaps in the set \( S \) are covered in part by the translation of \( S \) by the term of \( S \) just before this gap. Hence, at least the fraction \( \alpha \) of this gap gets covered. So from this covering we have density \( \alpha \) from \( S \) itself and \( \alpha \) times the gaps. Altogether, th... | Yes |
Lemma 2. If \( S \) has density \( \alpha > \frac{1}{2} \), then \( S \oplus S \) contains all the positive integers. | Proof. Fix an integer \( n \) which is arbitrary, let \( A \) be the subset of \( S \) which lies \( \leq n \), and let \( B \) be the set of all \( n \) minus elements of \( S \) . Since \( A \) contains more than \( n/2 \) elements and \( B \) contains at least \( n/2 \) elements, the Pigeonhole principle guarantees ... | Yes |
Lemma 3. Let \( k > 1 \) be a fixed integer. There exists a \( {C}_{1} \) such that, for any positive integers \( N, a, b \) with \( \left( {a, b}\right) = 1 \) , \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {\frac{a}{b}{n}^{k}}\right) }\right| \leq {C}_{1}{N}^{1 + o\left( 1\right) }{b}^{-{2}^{1 - k}}. \] | Null | No |
Lemma 4. There exists \( \epsilon > 0 \) and \( {C}_{2} \) such that, throughout any interval \( {I}_{a, b, N} \) , \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {x{n}^{k}}\right) }\right| \leq \frac{{C}_{2}N}{{\left( b + j\right) }^{\epsilon }} \] | Proof. This is almost trivial if \( b > {N}^{2/3} \), for, since the derivative of \( \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {x{n}^{k}}\right) }\right| \) is bounded by \( {2\pi }{N}^{k + 1} \) , \[ \left| {\mathop{\sum }\limits_{{n = 1}}^{N}e\left( {x{n}^{k}}\right) }\right| \leq \left| {\mathop{\sum }\lim... | Yes |
Theorem 1.1.4 Let \( {A}_{0},{A}_{1},{A}_{2},\ldots \) be countable sets. Then their union \( A = \mathop{\bigcup }\limits_{0}^{\infty }{A}_{n} \) is countable. | Proof. For each \( n \), choose an enumeration \( {a}_{n0},{a}_{n1},{a}_{n2},\ldots \) of \( {A}_{n} \) . We enumerate \( A = \mathop{\bigcup }\limits_{n}{A}_{n} \) following the above diagonal method. | Yes |
Theorem 1.1.4 Let \( {A}_{0},{A}_{1},{A}_{2},\ldots \) be countable sets. Then their union \( A = \mathop{\bigcup }\limits_{0}^{\infty }{A}_{n} \) is countable. | Proof. For each \( n \), choose an enumeration \( {a}_{n0},{a}_{n1},{a}_{n2},\ldots \) of \( {A}_{n} \) . We enumerate \( A = \mathop{\bigcup }\limits_{n}{A}_{n} \) following the above diagonal method. | Yes |
Theorem 1.1.8 (Cantor) For any two real numbers \( a, b \) with \( a < b \), the interval \( \left\lbrack {a, b}\right\rbrack \) is uncountable. | Proof. (Cantor) Let \( \left( {a}_{n}\right) \) be a sequence in \( \left\lbrack {a, b}\right\rbrack \) . Define an increasing sequence \( \left( {b}_{n}\right) \) and a decreasing sequence \( \left( {c}_{n}\right) \) in \( \left\lbrack {a, b}\right\rbrack \) inductively as follows: Put \( {b}_{0} = a \) and \( {c}_{0}... | Yes |
Theorem 1.1.8 (Cantor) For any two real numbers \( a, b \) with \( a < b \), the interval \( \left\lbrack {a, b}\right\rbrack \) is uncountable. | Proof. (Cantor) Let \( \left( {a}_{n}\right) \) be a sequence in \( \left\lbrack {a, b}\right\rbrack \) . Define an increasing sequence \( \left( {b}_{n}\right) \) and a decreasing sequence \( \left( {c}_{n}\right) \) in \( \left\lbrack {a, b}\right\rbrack \) inductively as follows: Put \( {b}_{0} = a \) and \( {c}_{0}... | Yes |
Theorem 1.1.9 The set \( \{ 0,1{\} }^{\mathbb{N}} \), consisting of all sequences of 0’s and 1’s, is uncountable. | Proof. Let \( \left( {\alpha }_{n}\right) \) be a sequence in \( \{ 0,1{\} }^{\mathbb{N}} \) . Define \( \alpha \in \{ 0,1{\} }^{\mathbb{N}} \) by\n\n\[ \alpha \left( n\right) = 1 - {\alpha }_{n}\left( n\right), n \in \mathbb{N}. \]\n\nThen \( \alpha \neq {\alpha }_{i} \) for all \( i \) . Since \( \left( {\alpha }_{n}... | Yes |
Theorem 1.2.1 (Cantor) For any set \( X, X{ < }_{c}\mathcal{P}\left( X\right) \) . | Proof. First assume that \( X = \varnothing \) . Then \( \mathcal{P}\left( X\right) = \{ \varnothing \} \) . The only function on \( X \) is the empty function \( \varnothing \), which is not onto \( \{ \varnothing \} \) . This observation proves the result when \( X = \varnothing \) .\n\nNow assume that \( X \) is non... | Yes |
Theorem 1.2.3 (Schröder - Bernstein Theorem) For any two sets \( X \) and \( Y \) , \[ \left( {X{ \leq }_{c}Y\& Y{ \leq }_{c}X}\right) \Rightarrow X \equiv Y. \] | Proof. (Dedekind) Let \( X{ \leq }_{c}Y \) and \( Y{ \leq }_{c}X \) . Fix one-to-one maps \( f : X \rightarrow Y \) and \( g : Y \rightarrow X \) . We have to show that \( X \) and \( Y \) have the same cardinality; i.e., that there is a bijection \( h \) from \( X \) onto \( Y \) .\n\nWe first show that there is a set... | Yes |
Corollary 1.2.4 For sets \( A \) and \( B \) , | \[ A{ < }_{c}B \Leftrightarrow A{ \leq }_{c}B\& B{ \nleq }_{c}A. \] | Yes |
Example 1.2.5 Define \( f : \mathcal{P}\left( \mathbb{N}\right) \rightarrow \mathbb{R} \), the set of all real numbers, by\n\n\[ f\left( A\right) = \mathop{\sum }\limits_{{n \in A}}\frac{2}{{3}^{n + 1}}, A \subseteq \mathbb{N}. \]\n\nThen \( f \) is one-to-one. Therefore, \( \mathcal{P}\left( \mathbb{N}\right) { \leq }... | Now consider the map \( g \) :\n\n\( \mathbb{R} \rightarrow \mathcal{P}\left( \mathbb{Q}\right) \) by\n\n\[ g\left( x\right) = \{ r \in \mathbb{Q} \mid r < x\}, x \in \mathbb{R}. \]\n\nClearly, \( g \) is one-to-one and so \( \mathbb{R}{ \leq }_{c}\mathcal{P}\left( \mathbb{Q}\right) \) . As \( \mathbb{Q} \equiv \mathbb... | Yes |
Example 1.2.6 Fix a one-to-one map \( x \rightarrow \left( {{x}_{0},{x}_{1},{x}_{2},\ldots }\right) \) from \( \mathbb{R} \) onto \( \{ 0,1{\} }^{\mathbb{N}} \), the set of sequences of 0 ’s and 1’s. Then the function \( \left( {x, y}\right) \rightarrow \) \( \left( {{x}_{0},{y}_{0},{x}_{1},{y}_{1},\ldots }\right) \) f... | Null | No |
Theorem 1.3.1 If \( X \) is infinite and \( A \subseteq X \) finite, then \( X \smallsetminus A \) and \( X \) have the same cardinality. | Proof. Let \( A = \left\{ {{a}_{0},{a}_{1},\ldots ,{a}_{n}}\right\} \) with the \( {a}_{i} \) ’s distinct. By \( \mathbf{{AC}} \), there exist distinct elements \( {a}_{n + 1},{a}_{n + 2},\ldots \) in \( X \smallsetminus A \) . To see this, fix a choice function \( f : \mathcal{P}\left( X\right) \smallsetminus \{ \varn... | Yes |
Corollary 1.3.2 Show that for any infinite set \( X,\mathbb{N}{ \leq }_{c}X \) ; i.e., every infinite set \( X \) has a countable infinite subset. | Null | No |
Example 1.3.4 Let \( X \) and \( Y \) be any two sets. A partial function \( f \) : \( X \rightarrow Y \) is a function with domain a subset of \( X \) and range contained in \( Y \) . Let \( f : X \rightarrow Y \) and \( g : X \rightarrow Y \) be partial functions. We say that \( g \) extends \( f \), or \( f \) is a ... | Null | No |
Example 1.3.5 Let \( V \) be a vector space over any field \( F \) and \( P \) the set of all independent subsets of \( V \) ordered by the inclusion \( \subseteq \) . Then \( P \) is a poset that is not a linearly ordered set. | In 1.3.5, Let \( C \) be a chain in \( P \) . Then for any two elements \( E \) and \( F \) of \( P \), either \( E \subseteq F \) or \( F \subseteq E \) . It follows that \( \bigcup C \) itself is an independent set and so is an upper bound of \( C \) . | No |
Proposition 1.3.7 Every vector space \( V \) has a basis. | Proof. Let \( P \) be the poset defined in 1.3.5; i.e., \( P \) is the set of all independent subsets of \( V \) . Since every singleton set \( \{ v\}, v \neq 0 \), is an independent set, \( P \neq \varnothing \) . As shown earlier, every chain in \( P \) has an upper bound. Therefore, by Zorn’s lemma, \( P \) has a ma... | Yes |
Theorem 1.4.1 For any two sets \( X \) and \( Y \), at least one of\n\n\[ X{ \leq }_{c}Y\text{or}Y{ \leq }_{c}X \]\n\nholds. | Proof. Without loss of generality we can assume that both \( X \) and \( Y \) are nonempty. We need to show that either there exists a one-to-one map \( f : X \rightarrow Y \) or there exists a one-to-one map \( g : Y \rightarrow X \) . To show this, consider the poset \( {Fn}\left( {X, Y}\right) \) of all one-to-one p... | Yes |
Corollary 1.4.2 Let \( A \) and \( B \) be any two sets. Then exactly one of\n\n\[ A{ < }_{c}B, A \equiv B,\text{ and }B{ < }_{c}A \] \n\nholds. | Null | No |
Theorem 1.4.3 For every infinite set \( X \) ,\n\n\[ X \times \{ 0,1\} \equiv X \] | Proof. Let\n\n\[ P = \{ \left( {A, f}\right) : A \subseteq X\\text{ and }f : A \times \{ 0,1\} \rightarrow A\\text{ a bijection }\} .\n\]\n\nSince \( X \) is infinite, it contains a countably infinite set, say \( D \) . By 1.1.3, \( D \times \{ 0,1\} \equiv D \) . Therefore, \( P \) is nonempty. Consider the partial or... | Yes |
Theorem 1.4.5 For every infinite set \( X \) , \[ X \times X \equiv X. \] | Proof. Let \[ P = \{ \left( {A, f}\right) : A \subseteq X\text{ and }f : A \times A \rightarrow A\text{ a bijection }\} . \] Note that \( P \) is nonempty. Consider the partial order \( \propto \) on \( P \) defined by \[ \left( {A, f}\right) \propto \left( {B, g}\right) \Leftrightarrow A \subseteq B\& f \preccurlyeq g... | Yes |
Theorem 1.4.5 For every infinite set \( X \) , \[ X \times X \equiv X. \] | Proof. Let \[ P = \{ \left( {A, f}\right) : A \subseteq X\text{ and }f : A \times A \rightarrow A\text{ a bijection }\} . \] Note that \( P \) is nonempty. Consider the partial order \( \propto \) on \( P \) defined by \[ \left( {A, f}\right) \propto \left( {B, g}\right) \Leftrightarrow A \subseteq B\& f \preccurlyeq g... | Yes |
Proposition 1.4.8 (J. König,[58]) Let \( \\left\\{ {{X}_{i} : i \\in I}\\right\\} \) and \( \\left\\{ {{Y}_{i} : i \\in I}\\right\\} \) be families of sets such that \( {X}_{i}{ < }_{c}{Y}_{i} \) for each \( i \\in I \) . Then there is no map \( f \) from \( \\mathop{\\bigcup }\\limits_{i}{X}_{i} \) onto \( {\\Pi }_{i}... | Proof. Let \( f : \\mathop{\\bigcup }\\limits_{i}{X}_{i} \\rightarrow {\\Pi }_{i}{Y}_{i} \) be any map. For any \( i \\in I \\), let\n\n\[ \n{A}_{i} = {Y}_{i} \\smallsetminus {\\pi }_{i}\\left( {f\\left( {X}_{i}\\right) }\\right)\n\]\n\nwhere \( {\\pi }_{i} : \\mathop{\\prod }\\limits_{j}{Y}_{j} \\rightarrow {Y}_{i} \)... | Yes |
\[ {2}^{\mathfrak{c}} \leq {\aleph }_{0}^{\mathfrak{c}} \] | \[ {2}^{\mathfrak{c}} \leq {\aleph }_{0}^{\mathfrak{c}}\;\text{ (since }2 \leq {\aleph }_{0}\text{ ) } \] \[ \leq {\mathfrak{c}}^{\mathfrak{c}}\;\text{ (since }{\aleph }_{0} \leq \mathfrak{c}\text{ ) } \] \[ = {\left( {2}^{{\aleph }_{0}}\right) }^{\mathfrak{c}}\;\left( {\text{since}\mathfrak{c} = {2}^{{\aleph }_{0}}}\r... | Yes |
Proposition 1.6.2 A linearly ordered set \( \\left( {W, \\leq }\\right) \) is well-ordered if and only if there is no descending sequence \( {w}_{0} > {w}_{1} > {w}_{2} > \\cdots \) in \( W \) . | Proof. Let \( W \) be not well-ordered. Then there is a nonempty subset \( A \) of \( W \) not having a least element. Choose any \( {w}_{0} \\in A \) . Since \( {w}_{0} \) is not the first element of \( A \), there is a \( {w}_{1} \\in A \) such that \( {w}_{1} < {w}_{0} \) . Since \( {w}_{1} \) is not the first eleme... | Yes |
Example 1.6.3 Let \( W = \mathbb{N}\bigcup \{ \infty \} \). Let \( \leq \) be defined in the usual way on \( \mathbb{N} \) and let \( i < \infty \) for \( i \in \mathbb{N} \). Clearly, \( W \) is a well-ordered set. Since \( W \) has a last element and \( {\omega }_{0} \) does not, \( \left( {W, \leq }\right) \) is not... | Null | No |
Proposition 1.6.5 No well-ordered set \( W \) is order isomorphic to an initial segment \( W\left( u\right) \) of itself. | Proof. Let \( W \) be a well-ordered set and \( u \in W \) . Suppose \( W \) and \( W\left( u\right) \) are isomorphic. Let \( f : W \rightarrow W\left( u\right) \) be an order isomorphism. For \( n \in \mathbb{N} \) , let \( {w}_{n} = {f}^{n}\left( u\right) \) . Note that\n\n\[ \n{w}_{0} = {f}^{0}\left( u\right) = u >... | Yes |
Proposition 1.7.1 (Proof by induction) For each \( n \in \mathbb{N} \), let \( {P}_{n} \) be a mathematical proposition. Suppose \( {P}_{0} \) is true and for every \( n,{P}_{n + 1} \) is true whenever \( {P}_{n} \) is true. Then for every \( n,{P}_{n} \) is true. Symbolically, we can express this as follows. | The proof of this proposition uses two basic properties of the set of natural numbers. First, it is well-ordered by the usual order, and second, every nonzero element in it is a successor. A repeated application of 1.7.1 gives us the following. | No |
Proposition 1.7.2 (Definition by induction) Let \( X \) be any nonempty set. Suppose \( {x}_{0} \) is a fixed point of \( X \) and \( g : X \rightarrow X \) any map. Then there is a unique map \( f : \mathbb{N} \rightarrow X \) such that \( f\left( 0\right) = {x}_{0} \) and \( f\left( {n + 1}\right) = g\left( {f\left( ... | Null | No |
Theorem 1.7.3 (Proof by transfinite induction) Let \( \\left( {W, \\leq }\\right) \) be a well-ordered set, and for every \( w \\in W \), let \( {P}_{w} \) be a mathematical proposition. Suppose that for each \( w \\in W \), if \( {P}_{v} \) is true for each \( v < w \), then \( {P}_{w} \) is true. Then for every \( w ... | Proof. Let\n\n\[ \n\\left( {\\forall w \\in W}\\right) \\left( {\\left( {\\left( {\\forall v < w}\\right) {P}_{v}}\\right) \\Rightarrow {P}_{w}}\\right) .\n\]\n\n\( \\left( *\\right) \)\n\nSuppose \( {P}_{w} \) is false for some \( w \\in W \) . Consider\n\n\[ \nA = \\left\{ {w \\in W : {P}_{w}\\text{ does not hold }}\... | Yes |
Theorem 1.7.4 (Definition by transfinite induction) Let \( \\left( {W, \\leq }\\right) \) be a well-ordered set, \( X \) a set, and \( \\mathcal{F} \) the set of all maps with domain an initial segment of \( W \) and range contained in \( X \) . If \( G : \\mathcal{F} \\rightarrow X \) is any map, then there is a uniqu... | Proof. For each \( w \\in W \), let \( {P}_{w} \) be the proposition \ | No |
Theorem 1.7.5 (Trichotomy theorem for well-ordered sets) For any two well-ordered sets \( W \) and \( {W}^{\prime } \), exactly one of\n\n\[ W \prec {W}^{\prime }, W \sim {W}^{\prime },\text{ and }{W}^{\prime } \prec W \]\n\nholds. | Proof. It is easy to see that no two of these can hold simultaneously. For example, if \( W \sim {W}^{\prime } \) and \( {W}^{\prime } \prec W \), then \( W \) is isomorphic to an initial segment of itself. This is impossible by 1.6.5.\n\nTo show that at least one of these holds, take \( X = {W}^{\prime }\bigcup \{ \in... | Yes |
Corollary 1.7.6 Let \( \\left( {W, \\leq }\\right) ,\\left( {{W}^{\\prime },{ \\leq }^{\\prime }}\\right) \) be well-ordered sets. Then \( W \\preccurlyeq {W}^{\\prime } \) if and only if there is a one-to-one order-preserving map from \( W \) into \( {W}^{\\prime } \) . | Proof. Suppose there is a one-to-one order-preserving map \( g \) from \( W \) into \( {W}^{\\prime } \) . Let \( X \) and \( f : W \\rightarrow X \) be as in the proof of 1.7.5. Then, by induction on \( w \), we easily show that for every \( w \\in W, f\\left( w\\right) { \\leq }^{\\prime }g\\left( w\\right) \) . Ther... | Yes |
Theorem 1.7.7 Let \( \mathcal{W} = \left\{ {\left( {{W}_{i},{ \leq }_{i}}\right) : i \in I}\right\} \) be a family of pairwise non-isomorphic well-ordered sets. Then there is a \( W \in \mathcal{W} \) such that \( W \prec {W}^{\prime } \) for every \( {W}^{\prime } \in \mathcal{W} \) different from \( W \) . | Proof. Suppose no such \( W \) exists. Then there is a descending sequence\n\n\[ \cdots \prec {W}_{n} \prec \cdots \prec {W}_{1} \prec {W}_{0} \]\n\nin \( \mathcal{W} \) . For \( n \in \mathbb{N} \), choose a \( {w}_{n}^{\prime } \in {W}_{n} \) such that \( {W}_{n + 1} \sim {W}_{n}\left( {w}_{n}^{\prime }\right) \) . F... | Yes |
Theorem 1.8.3 Every ordinal \( \alpha \) can be uniquely written as \[ \alpha = \beta + n \] where \( \beta \) is a limit ordinal and \( n \) finite. | Proof. Let \( \alpha \) be an ordinal number. We first show that there exists a limit ordinal \( \beta \) and an \( n \in \omega \) such that \( \alpha = \beta + n \) . Choose a well-ordered set \( W \) such that \( t\left( W\right) = \alpha \) . If \( W \) has no last element, then we take \( \beta = \alpha \) and \( ... | Yes |
Theorem 1.8.5 The set \( \Omega \) of all countable ordinals is uncountable. | Proof. Suppose \( \Omega \) is countable. Fix an enumeration \( {\alpha }_{0},{\alpha }_{1},\ldots \) of \( \Omega \) . Then\n\n\[ \alpha = \mathop{\sum }\limits_{n}{\alpha }_{n} + 1 \]\n\n is a countable ordinal strictly larger than each \( {\alpha }_{n} \) . This is a contradiction. So, \( \Omega \) is uncountable. | Yes |
Proposition 1.8.6 Let \( \alpha \) be a countable limit ordinal. Then there exist \( {\alpha }_{0} < {\alpha }_{1} < \cdots \) such that \( \sup \left\{ {{\alpha }_{n} : n \in \mathbb{N}}\right\} = \alpha \) . | Proof. Since \( \alpha \) is countable, \( \{ \beta \in \mathbf{{ON}} : \beta < \alpha \} \) is countable. Fix an enumeration \( \left\{ {{\beta }_{n} : n \in \mathbb{N}}\right\} \) of all ordinals less than \( \alpha \) . We now define a sequence of ordinals \( \left( {\alpha }_{n}\right) \) by induction on \( n \) . ... | Yes |
Proposition 1.10.7 (König’s infinity lemma, [57]) Let \( T \) be a finitely splitting, infinite tree on \( A \) . Then \( T \) is ill-founded. | Proof. Let \( T \) be a finitely splitting, infinite tree on \( A \) . Let \( \left( {a}_{0}\right) \) be a node of \( T \) with infinitely many extensions in \( T \) . Since \( T \) is finitely splitting (and \( e \in T),\{ a \in A : \left( a\right) \in T\} \) is finite. Further, \( T \) is infinite. So, \( \left( {a}... | Yes |
Example 1.10.3 The tree\n\n\\[ T = \\{ e\\} \\bigcup \\left\\{ {i{0}^{j} : j \\leq i, i \\in \\mathbb{N}}\\right\\} \\]\n\nis infinite and well-founded. | Null | No |
Example 1.10.4 Let \( T \) be a tree and \( u \) a node of \( T \) . The set\n\n\[ \n{T}_{u} = \\left\\{ {v \\in {A}^{ < \\mathbb{N}} : u \\hat{} v \\in T}\\right\\} \n\]\n\nforms a tree. (See Figure 1.4.) | Null | No |
Proposition 1.10.7 (König’s infinity lemma, [57]) Let \( T \) be a finitely splitting, infinite tree on \( A \) . Then \( T \) is ill-founded. | Proof. Let \( T \) be a finitely splitting, infinite tree on \( A \) . Let \( \left( {a}_{0}\right) \) be a node of \( T \) with infinitely many extensions in \( T \) . Since \( T \) is finitely splitting (and \( e \in T),\{ a \in A : \left( a\right) \in T\} \) is finite. Further, \( T \) is infinite. So, \( \left( {a}... | Yes |
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