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'˙ W S n Xf˙enC1g !R n. The differential of ' ı '1 C W Rn X 0 ! Rn X 0 at x is the linear map Rn ! Rn; 7! kxk2 2hx; ix kxk4 : For kxk D1 we obtain the reflection 7! 2hx; ix at the hyperplane orthogonal to x. Let U˙ D p1.S n X fenC1g/. The differential of '˙ yields a homeomorphism which is fibrewise linear and the diagram... |
ee that .Rj ; rj / is a pullback. The desired bundle map R is the composition of the Rj according to the ordering of N. This is sensible, since for each x 2 E only a finite number of Rj .x/ are different from x. The condition maxfuj .x/ j j 2 J g D1 shows that R is a map over r. If we apply the previous proof to princip... |
iversal bundles so that we need not rely on a special construction. (14.4.12) Theorem. A numerable G-principal bundle q W E ! B is universal if and only if E is contractible (as a space without group action). 348 Chapter 14. Bundles Proof. We know already that Milnor’s space EG is contractible. If p is universal, the G... |
ng x 2 b to .f .b/; f .x// 2 Ek.Rn/ Gk.Rn/Rn. (Verify that b 7! f .b/ is continuous.) This map is called the Gauss map of . The bundle k is numerable of finite type, as a bundle over a compact Hausdorff space, hence the induced bundle has the same properties. Standard constructions of linear algebra can be applied fibrew... |
s U.n/ ! U.n C 1/, A 7! are used to define U D colim U.n/, a topological group with the colimit topology. The inclusion of groups U.n/ ! U induces BU.n/ ! BU. If we compose a classifying map X ! BU.n/ with this map, we call the result X ! BU the stable classifying map. Bundles and ˚ a" are called stably equivalent, and ... |
set F .x/ D '0.x/f .x/ for x 2 U and F .x/ D 0 otherwise. (15.1.3) Proposition. Let M be a submanifold of N . A smooth function f W M ! R has a smooth extension F W N ! R. Proof. From the definition of a submanifold we obtain for each p 2 M an open neighbourhood U of p in N and a smooth retraction r W U ! U \ M . Hence ... |
ch chart .U; h; V / of N the subset h.U \ A/ has measure zero in Rn. A subset of Rn has measure zero if it can be covered by a countable number of cubes with arbitrarily small total volume. We use the fact that a diffeomorphism (in fact a C 1-map) sends sets of measure zero to sets of measure zero. An open (non-empty) ... |
1) ˇ W G ! C , g 7! gx has constant rank by equivariance. Hence there exists an open neighbourhood of e in G such that ˇ.U / is a submanifold of M . Since C is locally closed in the locally compact space M , the set C is locally compact and therefore ˇ W G ! C is an open map (see (1.8.6)). Hence there exists an open se... |
e positively related. We call M orientable, if M has an orienting atlas. An orientation of a manifold is represented by an orienting atlas; and two such define the same orientation if their union contains only positively related charts. If M is oriented by an orienting atlas, we call a chart positive with respect to the... |
nd the tangent vectors to the submanifold M . Since the tangent space of a vector space is canonically identified with the vector space, we can consider ˛ as Þ an inclusion. Let f W M ! N be an immersion. Then Tf is fibrewise injective. We pull back TN along f and obtain a fibrewise injective bundle morphism i W TM ! f TN... |
some neighbourhood of M n U and ˆ W M ! R2nC1 ˚ RN D Rq; x 7! .f .x/; ‰.x// is an embedding which coincides on V with f (up to composition with the inclusion R2nC1 Rq). For 2n < q 1 the image of is nowhere dense and for 2n 1 < q 1 the image of is nowhere dense (theorem of Sard). Therefore in each neighbourhood of w 2 S... |
to B in a 2 A if and only if a is a regular value of p ı f W f 1.Y / ! Y ! Rk. Proof. The space TbB is the kernel of Tbp. The composition of Taf W TaA ! TbM=TbB with the isomorphism TbM=TbB Š T0Rk induced by Tb W TbM ! T0Rk is Ta.p ı f /. Now we apply the above remark from linear algebra. (15.9.2) Proposition. Let f W ... |
on, the advantage is Þ that the handles are themselves n-dimensional manifolds. If M 0 arises from M by attaching a k-handle, 15.10.6 Elementary surgery. then @M 0 is obtained from @M by a process called elementary surgery. Let h W S k1 Dnk ! X be an embedding into an .n 1/-manifold with image U . Then X X U ı has a pi... |
ms 1. Let M be a smooth n-manifold with an orienting atlas. Then there exists a unique homological Z-orientation such that the local orientations in Hn.M; M X xI Z/ are mapped via positive charts to a standard generator of Hn.Rn; Rn X 0I Z/. Conversely, if M is Zoriented, then M has an orienting atlas which produces th... |
n exactly the x2K Ux where the Ux are pair-wise same manner. Let K be finite. Choose U D disjoint open neighbourhoods of x. We then have S L x2K Hn.Ux; Ux X x/ Š Hn.U; U X K/; Hn.Ux; Ux X x/ Š Z: The image z.Ux; x/ of z.M / is a generator, the local orientation determined by the fundamental class z.M /. The local degree... |
itney). In more abstract terms the product can also be obtained from the Eilenberg–Zilber chain equivalences as in the case of the homology product. We use the product structure to prove a powerful theorem (Leray–Hirsch) which says roughly that the cohomology of the total space of a fibration is a free module over the c... |
also consider situations where (6) or (5) do not hold. This is the reason for requiring (2) and (3) separately. For (5) it is required that the products are defined. For the convenience of the reader we also display the properties in a table and refer to the detailed description above. 17.2. Multiplicative Cohomology Th... |
F khi .B/ and b 2 F l hj .B/, then a Y b 2 F kCl hiCj .B/. Proof. Choose pre-images a0 2 hi .B; B k1/, b0 2 hl .B; B l1/. Then a0 b0 2 hiCj .B B; B k1 B [ B B l1/. The product a Y b is the image of a b under the diagonal d . A cellular approximation d 0 W B ! B B of the diagonal d sends B kCl1 into .B B/kCl1 and the l... |
e up to sign, provided e.n/.n/ D n.e.n// D ˙.n1/. The morphism n.†X/ is a homomorphism, since the group structures are also induced by the cogroup structure of †X. In order to check the commutativity one has to arrange and prove several things: (1) The Hurewicz homomorphisms commute with suspensions. (2) The structure ... |
se the data of (17.7.1) in order to rewrite the MV-sequence. We work with a ˙ F , y 7! .b0; y/ multiplicative cohomology theory. The embedding j W F ! Dn is an h-equivalence. Therefore we have isomorphisms ˙ W hk.X˙/ i ' ˙ Š hk.Dn ˙ F / j Š hk.F /: 426 Chapter 17. Cohomology The restriction of 'C gives us another isomo... |
additive and multiplicative. Under the obvious finiteness conditions (e.g., finite CW-complexes) additivity is not needed. Let .p; p0/ W .E; E 0/ ! B be a relative fibration over a CW-complex. A Thom class for p is an element t D t.p/ 2 hn.E; E0/ such that the restriction to each fibre tb 2 hn.Fb; F 0 b/. We apply the the... |
finite type. The sets Cn D f 12n 1; 2n C 1Œ are open and disjoint. Over Cn Cn the isomorphism the bundle is of finite type. By additivity, we have for C D hk.EjC; E0jC / Š hk.EjCn; E0jCn/. The Thom classes over Cn yield a unique Thom class over C . Now we use the same argument for Dn D f 12n; 2n C 2Œ, f 1n; n C 1Œ and th... |
N D F an isomorphism is induced (see (11.9.7)). The diagram H k.EI F / Yt H kCn.E; E0I F / ˛Š Hom.Hk.EI F /; F / .tZ/ ˛Š Hom.HkCn.E; E0I F /; F / is commutative (by property (6) in 18.1.1) where ˛ is the isomorphism of the universal coefficient theorem. Since Yt is an isomorphism, we conclude that tZ is an isomorphism. ... |
@ W H1.M; M X AI R/ Š QH0.M X AI R/; and the latter is a free R-module of rank j0.M X A/j 1. Þ (18.3.7) Example. H 2.RP 2I Z/ Š Z=2. This is not a free Z-module. Hence the projective plane cannot be embedded into S 3. (A similar proof shows that RP 2n Þ has no embedding into S 2nC1.) (18.3.8) Remark. From Alexander du... |
s Lhk ! hk into natural transformations. We define a coboundary operator Lı W Lhk.L/ ! LhkC1.K; L/ as follows. Let V L be open. Choose U V as an open neighbourhood of K. Then we map representing elements via ı W hk.V / ! hkC1.U; V /. This process yields a well-defined Lı and the diagram Lhk.L/ hk.K/ Lı ı LhkC1.K; L/ hkC1... |
val H n.M / ZŒM H0.M / " K. The bilinear form in question is isomorphic to the Hom-evaluation, and the latter is for each finite-dimensional vector space a regular form. 458 Chapter 18. Duality We take now R D K as coefficients and assume n D 4t. In that case H 2t .M / H 2t .M / ! R; .x; y/ 7! x ˇ y is a regular symmetri... |
is the local index. 3. The section s W S n ! TS n S n RnC1; x D .x0; : : : ; xn/ 7! .x; .x2 0 1; x0x1; : : : ; x0xn// has the transverse zeros .1; 0; : : : ; 0/ with index 1 and .1; 0; : : : ; 0/ with index .1/n. 4. Find a vector field on S 2n with a single zero (of index 2). 5. There exists a section without zeros if ... |
class t.1/ 2 H 2.E1; E0 1 / is mapped to the generator e.2/ under ', where e.2/ is defined by the relation he.2/; e2 i D1 (Kronecker pairing). The element t1 is the image of e.2/ under H 2.C; C X 0/ ! H 2.CP 1; CP 1 X CP 0/ ! H 2.CP 1/ and the fundamental class ŒCP 1 is mapped to e2 under H2.CP 1/ ! H2.CP 1; CP 1 X CP ... |
(19.2.2) and (19.1.5) we obtain an injective map ˇ W h.BU.n// ! h.BU.n 1/ BU.1// Š h.BU.n 1//ŒŒ.n/: This fact yields, by induction on n, the claimed injectivity. Elements of h.BU.n// are called universal h./-valued characteristic classes for n-dimensional complex vector bundles. Given c 2 h.BU.n// and a classifying map... |
s of vector bundles is a bundle map. If we apply this to the universal onedimensional bundle, then we see that this bundle map preserves the Thom class t1. On the other hand, if we restrict to t1 2 h1.RP 1; /, this bundle map is of degree 1 and changes the sign of t1. Since t1 corresponds under suspension to a unit of ... |
ata W A m ! .A ˝ A/ Š A ˝ A and " W A e ! R Š R define the dual coalgebra .A; ; "/ of the algebra. In the case of graded modules we take the graded dual; if A D .An j n 2 N0/, then the dual is .An D Hom.An; R/ j n 2 N0/. 484 Chapter 19. Characteristic Classes (19.6.3) Example. Let W H .X/ ! Hom.H.X/; R/ be the map in th... |
e B .n/ b.n/ .b1; : : : ; bn/ D I./.b1; : : : ; bn/: .ˇ1; : : : ; ˇn/ D †I./.ˇ1; : : : ; ˇn/; P The polynomial b.n/ only involves the variables b1; : : : ; bjI./j and is independent of n for n jI./j. We denote this stable version by b. The b form an R-basis of the symmetric polynomials in RŒˇ. In this sense we can writ... |
al complex line bundle, now considered as oriented bundle; see (15.6.6). By the multiplicativity of the v-classes, we have for the tangent bundle 2k of CP 2k the relation v.2k/ D v./2kC1 D .1 C p1./q1 C p1./2q2 C / 2kC1 D .1 C c2q1 C c4q2 C / 2kC1 where as usual H .CP 2kI R/ Š RŒc=.c2kC1/. Note that p1./ D c2, by (19.5... |
X=A/ D feg. From Hi .X; A/ D QHi .X=A/ and (20.1.2) we conclude i .X=A/ D 0 for i < n. Let n D 2. Since X and A are simply connected, 1.X; A; / D 0 and the diagram 2.X; A; / Š 2.X=A; / h H2.X; A/ Š Š H2.X=A/ shows that h is an isomorphism. 498 Chapter 20. Homology and Homotopy By induction we know that j .X; A; / D 0 f... |
) From this diagram we obtain a resulting diagram of chain groups ArC1 ı CrC1.Y / d 0 rC1 Cr .Z/ 'r Cr .Y / d d 0 r Cr1.Z/ 'r1 Cr1.Y / with ArC1 D HrC1.ZrC1; Zr / a free abelian group with a basis given by the Þ .r C 1/-cells, and induced by .F; f /. We now start from a diagram in which ArC1 is a free abelian group wit... |
p0 W X0 ! B0. Let q W Y ! Dt be the pullback of p, and similarly q0 W Y 0 ! S t1 and q0 W Y0 ! Dt X 0. Then ‰ W hi .Y; Y 0/ ! hi .X; X 0/ is an isomorphism for each homology theory. Proof. We have a commutative diagram hi .Y; Y 0/ .1/ hi .Y; Y0/ .3/ hi .Y X Y 0; Y0 X Y 0/ ‰ ‰ ‰ .5/ hi .X; X 0/ .2/ hi .X; X0/ .4/ hi .X ... |
ained in C. There exists a map W X ! K.2; 2/ which induces an isomorphism 2. We pull back the path fibration along X2 ' X PK.2; 2/ f K.2; 2/. Since 2 2 C, we have Hi .K.2; 1// 2 C for i > 0, by the general assumption in this section. Note that K D K.2; 1/ is the fibre of ' and f . The exact homotopy sequence of is used t... |
j D 2k 1. This theorem indicates that homotopy theory becomes simpler “over the rationals”. In the so-called rational homotopy theory one constructs algebraic models for the rationalized homotopy theory. For an exposition see [65]. We discuss an example. Consider the path fibration i ! Y K.Z; 2/ ' K.Z; 3/ ! X ! K.Z; 3/ ... |
nifold to a space, and the boundary relation is induced by manifolds with boundary. Several of our earlier applications of homology and homotopy can easily be obtained just from the existence of bordism homology, e.g., the Brouwer fixed point theorem, the generalized Jordan separation theorem and the component theorem, ... |
V M an n-dimensional submanifold with boundary. If f W M ! X is a map which sends M X V into A, then ŒM; f D ŒV; f jV in Nn.X; A/. Proof. Consider F W M I ! X, .x; t/ 7! f .x/. Then @.M I / D M @I and V 1 [ M 0 is a submanifold of @.M I / whose complement is mapped under F into A. The definition of the bordism relation ... |
air is exact. In order to derive this sequence, consider the MV-sequence for the triad .X [ CAI X [ CA X X; X [ CA X /. The excision isomorphism hn.X X U; A X U / Š hn.X; A/ holds, provided there exists a function W X ! Œ0; 1 with U 1.0/ and 1Œ0; 1Œ A, since under this assumption the canonical map .X XU /[C.AXU / ! X [... |
! B over B and M./ is defined to be the (unpointed) mapping cone of s . From this definition we see that a bundle map f W ! induces a pointed map M.f / W M./ ! M./. In the category of compactly generated spaces we have a canonical homeomorphism M. / Š M./ ^ M./. If is the trivial one-dimensional bundle over a point, thi... |
re simply connected. Thus it suffices to see that we have a homology isomorphism in the same range. Hj .†M SO.k// Hj .M SO.k C 1// Thom Thom Hj k.BSO.k// Hj k.BSO.k C 1// The map BSO.k/ ! BSO.k C 1/ is .k 1/-connected, hence the vertical maps are isomorphisms for j k k 1. These arguments show that we need 538 Chapter 21... |
, Demonstratio non nullarum insignium proprietatum quibus solida hedris planis indusa sunt praedita. Novi commentarii academiae scientiarum Petropolitanae 4 (1752/3), 140–160. Opera Mathematica Vd. 26, 94–108 (1758) 309 544 Bibliography [62] Euler, L., Elementa doctrinae solidorum. Novi commentarii academiae scientiaru... |
plane. Math. Ann. 36 (1890), 157–160. [149] Pommerenke, Ch., Boundary behaviour of conformal maps. Grundlehren Math. Wiss. 299, Berlin, Springer 1992. 251 [150] Poincaré, H., Sur la généralisation d’un théorème d’Euler relatif aux polyèdres. Compt. Rend. Acad. Sci. Paris 117 (1893), 144–145. 310 [151] Poincaré, H., An... |
lation value, 4 action diagonal, 17 effective, 17 free, 17 left, 17 proper, 329 properly discontinuous, 64 right, 17 transitive, 17 trivial, 17 weakly proper, 329 acyclic, 287, 498 additive invariant, 309 additivity axiom, 245 adjoint map, 38 adjunction space, 7 affinely independent, 198 Alexander duality, 446 Alexander... |
, 25 product, 25 path component, 26 path connected, 26 Plücker coordinates, 366 Poincaré duality, 446 point finite, 318 pointed homotopy, 31 homotopy equivalence, 31 map, 31 product, 31 space, 31 sum, 31 polyhedron, 199 Pontrjagin class, 480 Pontrjagin number, 492 564 Index Pontrjagin–Thom construction, 530 pre-spectrum... |
uting into the differential equation gives y dx . Hence it is c = λeλx and Definition (Characteristic equation). The characteristic equation of a (secondorder) differential equation ay + by + c = 0 is aλ2 + bλ + c = 0. In this case there are two solutions to the characteristic equation, giving (in principle) two complemen... |
of the cos wave is known as beating. This happens when the forcing frequency is close to the natural frequency. The wavelength of the sin function has order O( 1 ). As ∆ω → 0, the wavelength of the beating envelope → ∞ and we just have the initial linear growth. ∆ω ) and cos has wavelength O( 1 ω0 Mathematically, sinc... |
all t = 0; ∞ −∞ lim ε→0 D(t; ε) dt = 1. 41 5 Second-order differential equations IA Differential Equations So we can replace the force in our example by ID(t; ε), and then take the limit as ε → 0. For example, we can choose D(t; ε) = 1 √ ε π e−t2/ε2 D ε = 1 ε = 0.5 t This has height O(1/ε) and width O(ε). It can be check... |
∂f ∂y are the Cartesian components of the gradient of We write ds = ˆs ds, where |ˆs| = 1. Then Definition (Directional derivative). The directional derivative of f in the direction of ˆs is df ds df ds = ˆs · ∇f. = ˆs · ∇f. Definition (Gradient vector). The gradient vector ∇f is defined as the vector that satisfies Official... |
ty, write x = x0 + ξ, y = y0 + η. Then ˙ξ = f (x0 + ξ, y0 + η) ∂f ∂x = f (x0, y0) + ξ (x0) + η ∂f ∂y (x0) + O(ξ2, η2) So if ξ, η 1, ˙ξ ˙η = fx gx ξ η fy gy This is a linear system, and we can determine its character from the eigensolutions. Example. (Population dynamics - predator-prey system) Suppose that there are x ... |
a basis, this is equivalent to saying γi(t) = αi(γ(t)), γi(0) = 0 for all i and t ∈ I. By the general theory of ordinary differential equations, there is an interval I and a solution γ, and any two solutions agree on their common domain. However, we need to do a bit more for uniqueness, since all we know is that there i... |
nifold. It is then a Lie group under multiplication. Then we have gln(R) = Lie(GLn(R)) = TI GLn(R) = TI Mn ∼= Mn. If A, B ∈ GLn(R), then So LA(B) = AB. DLA|B(H) = AH as LA is linear. We claim that under the identification, if ξ, η ∈ gln(R) = Mn, then [ξ, η] = ξη − ηξ. 30 3 Lie groups III Differential Geometry Indeed, on ... |
y for combination. To check independence, we write I = (i1, · · · , ip) and let vI = vi1 ∧ · · · ∧ vip . Then suppose aI vI = 0 for aI ∈ R. For each I, we let J be the multi-index J = {1, · · · , n} \ I. So if I = I , then vI ∧ vJ = 0. So wedging with vJ gives I I αI vI ∧ vJ = aI vI ∧ vJ = 0. So aI = 0. So done by (ii)... |
p-form on the manifold to obtain the “volume” of the manifold. Definition (Differential form). We write Ωp(M ) = C∞(M, ΛpT ∗M ) = {p-forms on M }. An element of Ωp(M ) is known as a differential p-form. In particular, we have Ω0(M ) = C∞(M, R). In local coordinates x1, · · · , xn on U we can write ω ∈ Ωp(M ) as ω = i1<...... |
∗ω indeed represents some member of H p dR(M ). Let [ω] ∈ H p dR(N ). Then dω = 0. So d(F ∗ω) = F ∗(dω) = 0. dR(M ). So this map makes sense. So [F ∗ω] ∈ H p To see it is well-defined, if [ω] = [ω], then ω − ω = dσ for some σ. So F ∗ω − F ∗ω = d(F ∗σ). So [F ∗ω] = [F ∗ω]. (iv) Follows from the corresponding fact for pul... |
˜en) = det B ω(e1, · · · , en). So ˜e1, · · · , ˜en is oriented iff det B > 0. We now generalize this to manifolds, where we try to orient the tangent bundle smoothly. Definition (Orientation of a manifold). An orientation of a manifold M is defined to be an equivalence class of elements ω ∈ Ωn(M ) that are nowhere vanish... |
Hn. Then by definition, we have TaHn = Dera(C∞(Hn, R)). We let i∗ : TaHn → TaRn be given by i∗(X)(g) = X(g|Hn ) We claim that i∗ is an isomorphism. For injectivity, suppose i∗(X) = 0. If f ∈ C∞(Hn), then f extends to a smooth g on some neighbourhood U of Hn. Then X(f ) = X(g|Hn ) = i∗(X)(g) = 0. So X(f ) = 0 for all f .... |
Proof. Trace through the definitions. 69 7 De Rham’s theorem* III Differential Geometry Proposition. Let U ⊆ Rn is convex, then U : H p dR(U ) → H p ∞(U, R) is an isomorphism for all p. Proof. If p > 0, then both sides vanish. Otherwise, we check manually that I : H 0 ∞(U, R) is an isomorphism. dR(U ) → H 0 These two are... |
h connection dE. Then there is an induced connection dE∗ on E∗ given by requiring ds, ξ = dEs, ξ + s, dE∗ ξ, for s ∈ Ω0(E) and ξ ∈ Ω0(E∗). Here · , · denotes the natural pairing Ω0(E) × Ω0(E∗) → C∞(M, R). So once we have a connection on E, we have an induced connection on all tensor products of it. Christoffel symbols W... |
Now if we want this to work, then V has to be parallel along any curve, and in particular for lines {y = β} for β = 0. If we stare at it long enough, we figure out a necessary condition is ∇ ∂ ∂xi ∇ ∂ ∂xj = ∇ ∂ ∂xj ∇ ∂ ∂xi . So the failure of these to commute tells us the curvature. This definition in fact works for any... |
31 subgroup, 33 Lie subgroup, 33 linear connection, 72 compatible, 78 metric, 78 symmetric, 79 torsion, 79 torsion-free, 79 local coordinates, 6 locally isometric, 83 manifold, 6 orientable, 57 orientation, 57 with boundary, 61 maximal integral curve, 25 Mayer-Vietoris sequence, 55 metric connection, 78 morphism vecto... |
olds of Euclidean space as in Examples 1.2.3 and 1.2.5. This is because we can visualize curves and surfaces in R3. However, there are a few topics in the later chapters which require the more abstract Definition 1.4.2 even to say interesting things about extrinsic geometry. There is a generalization to these manifolds ... |
umptions on f and the definition of φ that U ∩ M = p ∈ U f (p) = 0 = p ∈ U and so φ(U ∩M ) = Ω ∩Rm ×{0}. Hence the diffeomorphism φ : U → Ω satisfies the requirements of part (ii). This proves Theorem 2.1.10. φ(p) ∈ Rm × {0} The next corollary relates the notion of a smooth map on a smooth submanifold as defined in the beg... |
(p). Since TpM and the kernel of df (p) are both m-dimensional linear subspaces of Rk, we deduce that TpM = ker df (p). This proves part (iii) and Theorem 2.2.3. Exercise 2.2.4. Let M ⊂ Rk be a smooth m-dimensional manifold and let pi ∈ M be a sequence that converges to a point p ∈ M . Let τi be a sequence of nonzero ... |
e topology and the restriction F |U : U → V is a diffeomorphism. Hence the P -open set U ∩ P is diffeomorphic to the open set Ω := {y ∈ Rm−n | (q, y) ∈ V } ⊂ Rm−n by the diffeomorphism φ : U ∩ P → Ω, defined by φ(p) := Ap for p ∈ U ∩ P , whose inverse is the smooth map ψ : Ω → U ∩ P given by ψ(y) = (F |U )−1(q, y) for y ∈ ... |
f M if and only if P is discrete, i.e. every p ∈ P has an M -open neighborhood U such that U ∩ P = {p}. (iii) P is an m-dimensional submanifold of M if and only if P is M -open. Example 2.3.7. Let S1 ⊂ R2 ∼= C be the unit circle and consider the map f : S1 → R2 given by f (x, y) := (x, xy). This map is a proper immersi... |
r t ∈ R and p ∈ M . Then Theorem 2.4.9 asserts that φt is smooth for every t ∈ R and that φs+t = φs ◦ φt, φ0 = id (2.4.5) for all s, t ∈ R. In particular, this implies that φt ◦ φ−t = φ−t ◦ φt = id. Hence φt is bijective and (φt)−1 = φ−t, so each φt is a diffeomorphism. Exercise 2.4.13. Let M ⊂ Rk be a smooth manifold. ... |
orphic to o(3). Remark 2.4.24. There is a linear map Rm×m → Vect(Rm) : ξ → Xξ which assigns to a matrix ξ ∈ gl(m, R) the linear vector field Xξ : Rm → Rm given by Xξ(x) := ξx for x ∈ Rm. This map preserves the Lie bracket, i.e. [Xξ, Xη] = X[ξ,η], and hence is a Lie algebra homomorphism. To understand the Lie bracket geo... |
y Xξ(g) := ξg ∈ TgG, g ∈ G. (2.5.8) By Theorem 2.4.7 there is an integral curve γ : (−ε, ε) → G satisfying ˙γ(t) = Xξ(γ(t)) = ξγ(t), γ(0) = 1l. By (2.5.5), the curve (−ε, ε) → Rn×n : t → exp(tξ) satisfies the same initial value problem and hence, by uniqueness, we have exp(tξ) = γ(t) ∈ G for all t ∈ R with |t| < ε. Now ... |
olds of G. Such subgroups are called Lie subgroups. We assume throughout that G ⊂ GL(n, R) is a Lie group with the Lie algebra g := Lie(G) = T1lG. Definition 2.5.26 (Lie subgroup). A subset H ⊂ G is called a Lie subgroup of G iff it is both a subgroup and a smooth submanifold of G. A useful general criterion is the Close... |
tation and pushforward observe that φ∗Y = d dt t=0 φ ◦ ψt ◦ φ−1, gηg−1 = d dt t=0 g exp(tη)g−1, where ψt denotes the flow of Y . To understand the correspondence between the Lie brackets recall that [X, Y ] = d dt t=0 (φt)∗Y, [ξ, η] = d dt t=0 exp(tξ)η exp(−tξ), where φt denotes the flow of X. We emphasize that the analo... |
concept of a smooth vector bundle in the extrinsic setting (§2.6.2). 2.6.1 Submersions Let M ⊂ Rk be a smooth m-manifold and N ⊂ R be a smooth n-manifold. A smooth map f : N → M is called a submersion iff its derivative df (q) : TqN → Tf (q)M is surjective for every q ∈ N . Figure 2.12: A local right inverse of a submer... |
), D(p) := [s1(p) · · · sn(p)] ∈ R×n. By Exercise 2.6.9, this implies Π(p) = D(p)(D(p)TD(p))−1D(p)T for every p ∈ U . Thus every p0 ∈ M has a neighborhood U such that the restriction of Π to U is smooth. This shows that (ii) implies (iii). We prove that (iii) implies (iv). Fix a point p0 ∈ M and choose a ba- sis v1, . ... |
Rm−n by η := A(x, y)ξ and η := A(x, y)ξ. Then ∂A ∂x (x, y) · ξ + ∂A ∂y (x, y) · η ξ = ∂A ∂x (x, y) · ξ + (x, y) · η ξ. ∂A ∂y The graphs of the matrices A(z) determine a subbundle E ⊂ Ω × Rm with the fibers Ez := (ξ, η) ∈ Rn × Rm−n | η = A(x, y)ξ for z = (x, y) ∈ Ω. This subbundle is the image of the restriction E|U := ... |
tion maps are holomorphic. Prove that the manifold topology is the quotient topology, i.e. if π : Cn+1 \ {0} → CPn denotes the obvious projection, then a subset U ⊂ CPn is open if and only if π−1(U ) is an open subset of Cn+1 \ {0}. Example 2.8.6. The real projective space RPn is the set RPn = ⊂ Rn+1 | is a 1-dimension... |
derivative of f at p ∈ M is a linear map df (p) : TpM → Rn, and the formula (2.8.8) reads x := φα(p). df (p)[α, ξ]p = d(f ◦ φ−1 α )(x)ξ, This formula also applies to maps defined on some open subset of M . In particular, with f = φα : Uα → Rm we have dφα(p)[α, ξ]p = ξ. Thus the map dφα(p) : TpM → Rm is the canonical vec... |
lued functions. If the map Xα : φα(Uα) → Rm is defined by (2.8.9), then Xα ◦ φ−1 α = (ξ1, . . . , ξm). The above notation is motivated by the observation that the derivative of a smooth function f : M → R in the direction of a vector field X on a coordinate patch Uα is given by LX f |Uα = m i=1 ξi ∂f ∂xi . Here the term ... |
M is paracompact W has a locally finite refinement {Wj}j∈J . By the axiom of choice there is a map such that J → I : j → ij W j ⊂ Vij ∀ j ∈ J. Since the collection {Wj}j∈J is locally finite, we have Ki := Wj = ij =i ij =i W j ⊂ Vi by Lemma 2.9.11. Since V i is compact so is Ki. Step 3. There is a partition of unity subord... |
λ2 , 0, √ λξ 1 + λ2 = √ 1 1 + λ2 x + √ λ 1 + λ2 y. 112 CHAPTER 2. FOUNDATIONS Similarly, if pν ∈ K2iν for all ν, there exists a subsequence such that the limit λ := limν→∞|f ev(pν)|−1|f odd(pν)| exists and, by (2.9.4), this implies lim ν→∞ f (pν) |f (pν)| = 0, √ λξ 1 + λ2 , 0, √ ξ 1 + λ2 = √ λ 1 + λ2 x + √ 1 1 + λ2 y. ... |
= dψ0(x, 1l)(x, gg−1)g for all x ∈ Ω0, x ∈ Rn, g ∈ G, and g ∈ TgG, and the fact that the derivative dψ0(x, 1l) is bijective for all x ∈ Ω0 (even for all x ∈ Ω). Thus we have proved that ψ0 : Ω0 × G → P is an injective immersion. Shrinking Ω0 further, if necessary, we may assume that Ω0 has a compact closure and that ψ ... |
of M is an m-dimensional real vector space and hence is isomorphic to Rm. Thus any two tangent spaces TpM and TqM are of course isomorphic to each other. While there is no canonical isomorphism from TpM to TqM we shall see that every smooth curve γ in M connecting p to q induces an isomorphism between the tangent space... |
r, ˙γ is a vector field along γ and ∇ ˙γ(t) = Π(γ(t))¨γ(t). Hence ˙γ is a parallel vector field along γ if and only if ¨γ(t) ⊥ Tγ(t)M for all t ∈ I. We will return to this observation in Chapter 4. In general, a vector field X along a smooth curve γ : I → M is parallel ˙X(t) is orthogonal to Tγ(t)M for every t and, by the... |
d. This is the content of the next lemma. Lemma 3.3.9. Let M ⊂ Rn be a smooth m-manifold. For p ∈ M and u ∈ TpM define the linear map hp(u) : TpM → TpM ⊥ by hp(u)v := hp(u, v) = dΠ(p)uv (3.3.4) for v ∈ TpM . Then the following holds. (i) The adjoint operator hp(u)∗ : TpM ⊥ → TpM is given by hp(u)∗w = dΠ(p)uw, w ∈ TpM ⊥.... |
M ) at (p, e) ∈ F(M ) is the direct sum T(p,e)F(M ) = H(p,e) ⊕ V(p,e) of the horizontal space H(p,e) := (v, hp(v)e) v ∈ TpM and the vertical space V(p,e) := {0} × L(Rm, TpM ). (3.4.5) (3.4.6) (ii) The vertical space V(p,e) at (p, e) ∈ F(M ) is the kernel of the linear map dπ(p, e) : T(p,e)F(M ) → TpM. (iii) A curve β :... |
ntact in the track M . But the track is not moving; hence the point of contact in the wheel is not moving. One may explain the paradox this way: the train is moving forward and the wheel is rotating around the axle. The velocity of a point on the wheel is the sum of these two velocities. When the point is on the bottom... |
) choose any t0 ∈ I and any orthogonal matrix Ψ0 ∈ O(n) such that Ψ0|Tγ(t0)M = Φ(t0) and define Ψ(t) : Rn → Rn by (3.5.2). This proves Lemma 3.5.19. 3.5. MOTIONS AND DEVELOPMENTS 155 Remark 3.5.20. The operations of reparametrization, inversion, and composition yield developments when applied to developments; i.e. if (Φ... |
ψ ∂xi (c(t)) + m i,j=1 ξi(t) ˙cj(t) ∂2ψ ∂xi∂xj (c(t)). (3.6.4) (3.6.5) We examine the projection ∇X(t) = Π(γ(t)) ˙X(t) of this vector onto the tangent space of M at γ(t). The first summand on the right in (3.6.5) is already tγ( )X(t)c(t)UΩψφ Mξ( )t 162 CHAPTER 3. THE LEVI-CIVITA CONNECTION tangent to M . For the second ... |
ample 3.7.5 (Fubini–Study metric). The complex projective space carries a natural Riemannian metric, defined as follows. Identify CPn with the quotient of the unit sphere S2n+1 ⊂ Cn+1 by the diagonal action of the circle S1, i.e. CPn = S2n+1/S1. Then the tangent space of CPn at the equivalence class [z] = [z0 : · · · : ... |
odesics parametrized proportional to the arclength. We follow the latter course, referring to the more general concept as a “reparametrized geodesic”. Thus a reparametrized geodesic need not be a geodesic. We assume throughout that M ⊂ Rn is a smooth m-manifold. Definition 4.1.1 (Length and energy). Let I = [a, b] ⊂ R b... |
γ(t) = 1 2 d dt | ˙γ(t)|2 . Hence the function I → R : t → | ˙γ(t)|2 is constant. Choose c ≥ 0 such that | ˙γ(t)| ≡ c. If c = 0, then γ(t) is constant and so γ(t) ≡ p = q. If c > 0, then | ˙γs(t)| dt | ˙γs(t)| dt dL(γ)X = = = = = a b s=0 ∂ ∂s s=0 d ds b a b | ˙γ(t)|−1 a b 1 c 1 c a dE(γ)X. ˙γ(t), ∂ ∂s s=0 ˙γs(t) dt ˙γ(... |
: M → Tp0M at p = p0 is the identity on Tp0M . Hence the Inverse Function Theorem 2.2.17 asserts that the map x|M : M → Tp0M is locally invertible near p0. Extending this inverse to a smooth map from Tp0M to Rn and composing it with the map y : M → Tp0M ⊥, we obtain a smooth map f : Tp0M → Tp0M ⊥ and an open neighborh... |
of M , and the restriction of the exponential map to Br(p) is a diffeomorphism from Br(p) to Ur(p). Proof. This follows directly from Corollary 4.3.7 and Theorem 2.2.17. Definition 4.3.9 (Injectivity radius). Let M ⊂ Rn be a smooth mmanifold. The injectivity radius of M at p is the supremum of all real numbers r > 0 suc... |
it follows from Lemma 4.4.5 that | ˙γ(t)|2 = ˙β(t)2 ∂α ∂s (β(t), t) 2 + ∂α ∂t 2 (β(t), t) ≥ ˙β(t)2ε2 for every t ∈ I. Hence L(γ) = 1 0 | ˙γ(t)| dt = I | ˙γ(t)| dt ≥ ε I ˙β(t) dt ≥ ε I ˙β(t) dt = ε. 202 CHAPTER 4. GEODESICS Here the last equality follows by applying the fundamental theorem of calculus to each interval i... |
. Since it sends the pair (p0, 0) ∈ V to expp0(0) = p0 ∈ U , it follows from continuity that there exist constants ε > 0 and r > 0 such that p ∈ Ur, v ∈ TpM, |v| < ε =⇒ v ∈ Vp, expp(v) ∈ U. (4.5.6) Moreover, we have d expp0(0) = id : Tp0M → Tp0M by Corollary 4.3.7. Hence the Implicit Function Theorem 2.6.15 asserts tha... |
M \ expp(v) | v ∈ TpM, |v| ≤ ε . This shows that Sε(p) = expp(εΣ1(p)) and, since ε is smaller than the injectivity radius, the map Σ1(p) → Sε(p) : v → expp(εv) is a diffeomorphism. 4.6. COMPLETENESS AND HOPF–RINOW 213 To prove the second assertion, let q ∈ M such that r := d(p, q) > ε. Fix a constant δ > 0 and choose a... |
M → TpM as in Corollary 4.3.7. This leads directly to the injectivity radius, the Gauß Lemma 4.4.5, the local length minimizing property of geodesics in Theorem 4.4.4, and the Convex Neighborhood Theorem 4.5.3. Also the proof of the equivalence of metric and geodesic completeness in Theorem 4.6.5 and of the Hopf–Rinow ... |
oduct. It follows from the equation (4.6.3) in the proof of Lemma 4.6.8 that |v − w| = lim t→0 d(expp(tv), expp(tw)) t = lim t→0 = lim t→0 = lim t→0 d(φ(expp(tv)), φ(expp(tw))) t φ(p)(Φp(tv)), exp d(exp φ(p)(Φp(tw))) t d(exp φ(p)(tΦp(v)), exp φ(p)(tΦp(w))) t = |Φp(v) − Φp(w)| . 228 CHAPTER 5. CURVATURE Here the second ... |
finition. Definition 5.2.3 (Covariant derivative). Let M ⊂ Rn be an m-dimensional submanifold and X be a vector field on M . Fix a point p ∈ M and a tangent vector v ∈ TpM . The covariant derivative of X at p in the direction v is the tangent vector ∇vX(p) := Π(p)dX(p)v ∈ TpM, where Π(p) ∈ Rn×n denotes the orthogonal proj... |
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