class # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS

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1 class # MATH 7711, AUTUMN 2017 M-W-F 3:00 p.m., BE 128 A DAY-BY-DAY LIST OF TOPICS [DG] stands for Differential Geometry at [DFT] for Distributions and the Frobenius Theorem at [CC] for Metrics of Constant Curvature at [CF] for Conformal Flatness at [AC] for Algebraic Curvature Tensors at [SB] for Consequences of the Second Bianchi Identity at [RH] for Ricci-Hessian Equations at [EM] for Einstein Metrics in Dimension Four at August 23: Complete integrability of a system of first-order partial differential equations. The Frobenius Theorem stated in traditional terms. A fibration or (locally trivial) bundle pr : E B with the fibre F, defined by requiring that the base B be the union of open subsets U admitting local trivializations, that is, diffeomorphic identifications pr 1 (U) U F which make pr correspond to the product projection pr U : U F U. Sections and local sections of a fibration. Bundles with fibre geometry: vector and affine bundles, Riemannian fibre metrics, including Euclidean vector and affine bundles. Vector-bundle morphisms. References: [DFT, from formula (11) to formula (13)]]. August 25: An alternative description of vector bundles, based on atlases of systems of local trivializing sections. References: [DG, Section 16]. August 28: Vector subbundles (with the inclusion morphism). Their quotient vector bundles (with the projection morphism). The normal bundle of a vector subbundle. Distributions on a manifold M, that is, vector subbundles of the tangent bundle TM. The normal bundle D nrm = (TM)/D of a distribution D on M, with the quotient projection π : TM D nrm, and the curvature (tensor, form) of D, which is the vector-bundle morphism Ω : D 2 D nrm such that Ω(v, w) = π[v, w] for local sections v, w of D. References: [DG, Section 19]. August 30: More on tangent vectors, vector fields, and the Lie bracket [DG, Sections 5 6]. The reason why the curvature form Ω of a distribution is well defined. Integral manifolds of a distribution D on M, meaning: submanifolds P of M with T x P = D x 1

2 for all x P. Integrable distributions on M, defined by requiring that every point of M lie in an integral manifold. Flat connections, that is, integrable horizontal distributions in bundles. The Frobenius Theorem: Integrability of a distribution is equivalent to vanishing of its curvature. September 1: Some particularly interesting distributions on the total space E of a bundle pr : E B, namely, the vertical distribution V, with V y = Ker dpr y for all y E, and the class of horizontal distributions, also referred to as nonlinear connections in the bundle, that is, vector subbundles H of TE for which TE = H V. Integrability of the vertical distribution of a fibration pr : E B, with the fibres E x = pr 1 (x) serving as integral manifolds. Projectability of vector fields under smooth mappings and its relation with the Lie-bracket operation [DG, Section 6], including the fact that whenever two vector fields are tangent to a submanifold, so is their Lie bracket. The resulting proof of the easy part of the Frobenius Theorem (integrability implies Ω = 0). A stronger version of the Frobenius Theorem: Integrability of a distribution is equivalent to vanishing of its curvature, and a distribution is integrable if and only if it is, locally, the vertical distribution of a fibration. September 6: The initial step toward the proof of the difficult part of the Frobenius Theorem, consisting in a choice of local coordinates in the given m-dimensional manifold M such that our p-dimensional distribution D is complementary to the span of the last m p coordinate vector fields and thus, locally, constitutes a connection in a bundle. The corresponding equations for D (the distribution) and Ω, its curvature form [DFT, formulae (3), (6) (8)]. September 8: Submanifolds N of a manifold M (or, mappings ϕ : Q M from a manifold Q) tangent to a distribution D on M, in the sense that T x N D x (or, respectively, dϕ z (T z Q) D x ) whenever x N (or, z Q). A characterization of these objects in terms of solutions to a specific system of first-order partial differential equations (with D treated as a horizontal distribution for a standard fibration of a Euclidean rectangle). Proof of the first part of the Frobenius Theorem: using induction on the dimension of the base rectangle, one extends such solutions to rectangles of successive dimensions, starting from dimension one, where the existence and uniqueness (of horizontal lifts of curves in the base) are due to what we know about ordinary differential equations. September 11: Proof of the remaining part of the Frobenius Theorem: the induction step, establishing the first-order partial differential equations on a rectangle of the next dimension via the fact that zero is the only solution, assuming the value 0 somewhere, of a system of linear homogeneous ordinary differential equations. The stronger version of the Frobenius Theorem, derived from the inverse mapping theorem. September 13: More on submersions, immersions, and submanifolds. The continuousversus-smooth lemma [DG, Lemma 9.3], and the resulting uniqueness of the manifold structure of a submanifold with the subset topology [DG, Corollary 9.5]. Leaves of a distribution, defined to be its maximal connected integral manifolds. The fact that the intersection of two integral manifolds of any (possibly nonintegrable) distribution is an 2

3 open subset of both, relative to their manifold structures. September 15: The Leaf Theorem (without the assumption of integrability). The Leaf-Mapping Theorem for an integrable distribution, and the resulting uniqueness of the manifold structures of its integral manifolds. Lie groups and left-invariant vector fields [DG, Sections 4 and 8]. September 18: Smoothness of left-invariant vector fields, the Lie algebra g of a Lie group G, and the evaluation isomorphisms g T x G, for x G [DG, Section 8]. Liegroup homomorphisms ϕ : G H, constancy of their rank, the unique projectability of left-invariant vector fields on G onto left-invariant vector fields on H, and the resulting Lie-algebra homomorphism ϕ : g h [DG, Section 8]. Example: det = tr, where G = GL(V ) and H is the multiplicative group of nonzero (real or complex) scalars [DG, Example 8.9]. Functorial dependence of ϕ on ϕ [DG, Section 8]. Lie subgroups, and the Lie-subalgebra inclusion h g for a Lie subgroup H of a Lie group G [DG, Section 11]. Kernels of Lie-group homomorphisms as examples of (normal) Lie subgroups with the subset topology. Left-invariant distributions on a Lie group G, and the canonical bijective correspondence between them and the vector subspaces of g (where g always denotes the Lie algebra of G, defined to be the space of left-invariant vector fields on G). The dichotomy effect: a left-invariant distribution on G is integrable if the corresponding vector subspace is a Lie subalgebra, and has no integral manifolds otherwise. September 20: The fact that a constant-rank mapping (such as a Lie-group homomorphism), if injective, must be an immersion. The Lie-Subgroup Theorem and the Image- Group Theorem. September 22: The Koszul definition of a linear connection in a real/complex vector bundle pr : E M. The standard flat connection, denoted here by D, in a product vector bundle pr M : M F M, with D v ψ = d v ψ for smooth sections ψ treated as functions M F. Abundance of cut-off functions [DG, Section 83]. The local character of (meaning that the restriction of v ψ to an open set U depends only on the restrictions of v and ψ to U). The resulting restrictibility of to a connection in the restriction vector bundle pr : pr 1 (U) U, for any open submanifold U of M, that is, well-definedness of v ψ when v and ψ are only smooth local sections of TM and E. The component functions Γja b of relative to a local coordinate system xj in M and local trivializing sections e a of E, with j e a = Γja be b, and the formula vψ = v j ( j ψ a +Γjb aψb )e a for smooth local sections v of TM and ψ of E. The conclusion that the dependence of v ψ on v is pointwise (and not just local): at any x M, the value ( v ψ) x E x is uniquely determined by v x (and by the restriction of ψ to any given neighborhood of x). Well-definedness of w ψ E x for w T x M and a smooth local section of E defined on a neighborhood of x. The resulting linear operator ( ψ) x : T x M E x, for any such ψ, and the existence of such ψ with ( ψ) x = 0 having any prescribed value ψ x E x. The interpretation of ψ as a section of Hom(TM, E) (that is, a vector-bundle morphism TM E, cf. [DG, 3

4 Sections 18 and 23]) whenever ψ is a smooth section of a vector bundle pr : E M with a fixed linear connection. The comma notation ψ a,j instead of ( ψ) a j (which one also writes as j ψ a ), for local smooth sections ψ, so that ψ a,j = j ψ a + Γjb aψb. The linear operator T x M w w hrz φ T (x,φ) E of horizontal lift associated with, any x M, and any φ E x, given by w hrz φ = (dψ) x w, where ψ is any smooth local section of E defined on a neighborhood of x and having ( ψ) x = 0. Correctness of this definition (that is, its independence of the choice of ψ): as ψ a (x) = φ a and ( j ψ a )(x) = Γjb a(x)φb, in the local coordinates x j, φ a for E arising from our local coordinate system x j in M and local trivializing sections e a of E (see [DG, Section 17]), the components of (dψ) x w consist of w j (the components of w relative to x j ) and w j ( j ψ a )(x) = w j Γjb a(x)φb. The relation dpr (x,φ) w hrz φ = w, immediate from the chain rule [DG, formula (5.18)], applied to the equality pr ψ = id (and reflected by the just-mentioned fact that the initial components of w hrz φ are w j, the components of w). September 25: The horizontal distribution H on E corresponding to, with H y = {w hrz φ : w T x M} for y = (x, φ) E. The vertical and horizontal projection bundle morphisms TE = H V V and TE H, depending on via H, and written as [ ] vrt, [ ] hrz. The horizontal lift of a (local) vector field v tangent to M, relative to a linear connection in a vector bundle pr : E M, defined to be the (local) vector field v hrz tangent to E with (v hrz ) y = w hrz φ whenever y = (x, φ) E, so that x = pr(y), and w = v x. The components v j, v k Γkb aφb of v hrz in local coordinates x j, φ a for E discussed earlier, which we informally express as v hrz (v j, v k Γkb aφb ), and the resulting relations ξ hrz (ξ j, Γkb aφb ξ k ), ξ vrt (0, ξ a + Γkb aφb ξ k ) whenever ξ is a vector (field) tangent to the total space E, with ξ (ξ j, ξ a ), and ξ vrt, ξ hrz are its V and H components (projections). Smoothness of H. The equality ( w ψ) x = [(dψ) x w] vrt, for any x M, any w T x M and any smooth local section ψ of E defined on a neighborhood of x, showing that H uniquely determines. Proof of the last equality based on noting that, in local coordinates x j, φ a for E described above, (dψ) x w (w j, w k ( k ψ a )(x)), which makes it the sum of the horizontal vector w hrz φ (w j, w k Γkb a(x)ψb (x)) and the vertical vector with the components (0, w k [( k ψ a )(x) + Γkb a(x)ψb (x)]), that is, the vertical vector identified with ( w ψ) x. The equality ( w ψ) x = [(dψ) x w] vrt as a definition of the covariant derivative for arbitrary connections in arbitrary fibre bundles. Reference: [DG, Section 20]. September 27: The fact that a C 1 mapping ϕ between finite-dimensional real/complex vector spaces, with ϕ(εx) = εϕ(x) for all x in the domain space and some scalar ε / {0, 1}, is necessarily linear. The conclusion that a horizontal distribution in a vector bundle comes from a linear connection if and only if it is invariant under the multiplication by some/every scalar ε / {0, 1}. The curvature tensor R = R of a linear connection in a vector bundle pr : E M, assigning to x M the skew-symmetric bilinear mapping 4

5 R x = T x M T x M EndE x, and characterized by R(v, w)ψ = w v ψ v w ψ+ [v,w] ψ for smooth local sections v, w of TM and ψ of E. The pointwise dependence of R(v, w)ψ on v, w and ψ, due to the easily-verified formula [R(v, w)ψ] a = R a jkb v j w k ψ b, where the component functions R a jkb of R relative to any x j and e a as above are given by R a jkb = k Γjb a j Γ kb a + Γ kc aγ jb c Γ jc aγ kb c. Reference: [DG, Section 20]. September 29: The equality ([v hrz, w hrz ] vrt ) y = R x (v x, w x )φ for y = (x, φ) E and vector fields v, w tangent to M, showing that R = R essentially becomes the curvature Ω of the horizontal distribution H on E corresponding to, as long as one identifies the normal bundle H nrm with the vertical distribution V, and the quotient projection π : TE H nrm with the vertical projection TE V. A proof of this last equality, based on the coordinate expression of the Lie bracket [DG, formula (6.7)], the fact that [v hrz, w hrz ] projects under pr onto [v, w] (cf. [DG, Theorem 6.1]), along with the following relations (see September 25): ξ vrt (0, ξ a + Γkb aφb ξ k ) (applied to ξ = [v hrz, w hrz ]) and v hrz (v j, v k Γkb aφb ), as well as the analog of the latter for w hrz. The functor Hom applied to linear connections in vector bundles [DG, Section 24], with the component formula Φ λ a,j = j Φλ a +Γjµ λφµ a Γja bφλ b and the local definition j (e a e λ ) = Γ µ jλ ea e µ Γjb aeb e λ for the resulting connection in Hom(E, E ), denoted by, where x j, e a, e λ are, respectively, a local coordinate system in the base manifold M, and local trivializing sections for the bundles E, E. Reference: [DG, Sections 6 and 24]. October 2: The dual connection in E = Hom(E, M IK), the dual bundle of E, where IK = IR or IK = C. The conjugate connection in the conjugate bundle E of a complex vector bundle. Linear connections on a manifold (that is, in its tangent bundle), the torsion tensor field and its component functions [DG, Section 21]. Torsion-free connections, also referred to as symmetric. Pseudo-Riemannian and Riemannian fibre metrics in real vector bundles. Hermitian fibre metrics in complex vector bundles. Pseudo-Riemannian and Riemannian metrics and manifolds. Compatibility of a fibre metric θ in a vector bundle with a linear connection, meaning that θ = 0, with θ viewed as a smooth section of Hom(E, E ). The Leibniz-rule form of compatibilty, and its component version: j θ ab = Γja cθ cb + Γ jb cθ ac. Reference: [DG, Sections 23, 24, 28]. October 4: The reciprocal of a fibre metric θ in a real vector bundle, which is a fibre metric in the dual bundle having the component functions θ ab with θ ac θ cb = δb a. Raising and lowering indices. The Levi-Civita connection of a pseudo-riemannian manifold (M, g), which is the unique torsion-free connection in TM, compatible with g. The Christoffel symbols Γjk l l given by Γjk = Γ jkq g ql and 2Γ jkl = j g kl + k g jl l g jk. Vanishing of the connection components at a point. The first Bianchi identity R q jkl + R q klj + R q ljk = 0 for torsion-free connections. The second covariant derivative with the component functions ψ a,jk = k ψ a,j + Γkb aψb,j Γkj l ψa,l. Reference: [DG, Sections 29, 30, 26 (including Problems 1 2), 24]. 5

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