DIFFERENTIAL GEOMETRY. PROBLEM LIST. 1. Manifolds (1) Give the definition of a topological manifold. Which of the following topo- logical spaces are manifolds: • Rn with standard topology, • Rn with discrete topology, • the cross f(x; y) 2 R2 j xy = 0g, • the cusp f(x; y) 2 R2 j x2 = y3g, and • the circle f(x; y) 2 R2 j x2 + y2 = 1g? (2) Give an example of a topological space that is not a topological manifold and that is • Hausdorff and second countable, • second countable and locally Euclidean, and • locally Euclidean and Hausdorff. (3) Give the definition of a smooth manifold. Which of the following topological manifolds is/are smooth manifolds: • the cusp C = f(x; y) 2 R2j x2 = y3g with the atlas given by one coordi- nate chart U = C ! R1,(x; y) ! x, • the sphere f(x; y; z) 2 R3 j x2 + y2 + y3 = 1g; • the graph of a smooth function, 2 • the general linear group GL(n; R) ⊂ Rn . (4) Show that the product of two smooth manifolds is a smooth manifold. (5) Give the definition of a smooth map at a point. Give the definition of a smooth map of manifolds. • Show that if a continuous map F : M ! N of smooth manifolds is C1 at a point p 2 M with respect to a coordinate neighborhood about p and a coordinate neighborhood about F (p), then F is C1 at p with respect to any pair of coordinate neighborhoods about p and F (p). • Show that the composition of smooth maps is a smooth map. (6) Give the definition of a diffeomorphism of two manifolds. Let R be the real line with the standard smooth structure. Let R0 denote the real line with the differentiable structure given by the maximal atlas of the coordinate chart : R ! R, (x) = x1=3. Show that • the differentiable structures on R and R0 are distinct. • R is diffeomorphic to R0. (7) Show that the projection M × N ! M of the product of two manifolds is a smooth map. (8) Show that the inclusion M ! M × N, given by x 7! (x; y) where y is a fixed point in N, is a smooth map. More generally, show that given smooth manifolds M; K and L, a map (f; g): M ! K × L is smooth if and only if both f and g are smooth. 1 2 DIFFERENTIAL GEOMETRY. PROBLEM LIST. 2. Quotients and Tangent spaces (1) Let ∼ be an equivalence relation on a topological space S. Give the definition of the quotient topology on S=∼. • Prove that the quotient topology is indeed a topology. • Let π : S ! S=∼ denote the quotient map. Show that a map f : S=∼ ! Y to a topological space is continuous if and only if f ◦ π is continuous. (2) Give the definition of an open equivalence relation on a topological space S. Suppose that ∼ is an open relation on S. • Show that S=∼ is Hausdorff if and only if the graph R = f(x; y) 2 S × S j x ∼ yg is closed in S × S. • Show that if S is second countable, the S=∼ is also second countable. (3) Suppose a right action S × G ! S of a topological group G on a topological space S is continuous. Define two points x; y of S to be equivalent if there is an element g 2 G such that y = xg. Let S=G be the quotient space; it is called the orbit space of the action. Prove that the projection map π : S ! S=G is an open map. (4) Give the definition of RP n. Show that RP n is a topological manifold and introduce a smooth structure on RP n. (5) Give the definition of a derivation at a point in Rn. Show that the vector space of derivations at a point p is isomorphic to the vector space of vectors at p. (6) Give the definition of a derivation at a point in a manifold M. Give the definition of a tangent vector, and the tangent space TpM of M at p. Show that TpM is a vector space. Give the definition of the differential df : TpM ! Tf(p)N of a map f : M ! N. Show that the differential of a map is well- defined, i.e., it takes a derivation at a point in M to a derivation at a point in N. (7) Prove the chain rule for differentials of maps. Prove that coordinate vectors at a point in a manifold form a basis for the tangent space TpM. (8) Give the definition of a Lie group. Let G be a Lie group with multiplication map µ: G × G ! G, inverse map i : G ! G, and identity element e. • Show that de;eµ(X; Y ) = X + Y . • Show that dei(X) = −X. (9) Given a smooth map F : M ! N and a point p 2 M, let (U; x1; :::; xm) de- note a coordinate neighborhood in M about p, and (V; y1; :::; yn) a coordinate neighborhood in N about F (p). Relative to the coordinate bases f@=@xig for TpM and f@=@yjg for TF (x)N, find the matrix representing the differential 2 3 dpF . Compute the differential df of the map f : R ! R given by f(x; y) = (x2 + 3y; xy + 2x; y3 + x2) at any point (x; y), and find df(@x). DIFFERENTIAL GEOMETRY. PROBLEM LIST. 3 3. Submanifolds (1) Give the definition of a regular submanifold. Show that a regular subman- ifold has a structure of a smooth manifold. Which of the following topological subspaces are regular submanifolds. • the circle S1 = f(x; y) 2 R2 j x2 + y2 = 1 g in R2, • the closed interval [0; 1] ⊂ R2, • the open interval (0; 1) ⊂ R2, • the union of the graph of the topologists' sine curve y = sin(1=x) and the interval f0g × (−1; 1). (2) Give the definition of a local diffeomorphism. State the inverse function theorem for Rn and deduce the inverse function theorem for manifolds. Prove the corollary: let N be a smooth manifold of dimension n. Smooth functions F1; ::; Fn on a coordinate neighborhood (U; x1; :::; xn) of a point p in N define a coordinate neighborhood (U; F1; ::; Fn) if and only if the Jacobian determinant det(@Fi=@xj) 6= 0. (3) Give the definition of a critical point and regular point of a smooth map. Give the definition of a critical value and regular value. Find critical/regular points and critical/regular values of the wrinkling map: w : Ry × Rx × Rz −! Rt1 × Rt2 ; w(y; x; z) = (y; z3 + 3(y2 − 1)z + x2): (4) Prove the regular level set theorem: Let c be a regular value of a smooth map f : M ! N of manifolds. If f −1(c) is non-empty, then it is a regular submanifold of M of dimension dim M − dim N. (5) Let N ⊂ R3 be the set of solutions to the system of equations 4x2+9y2+z2 = 1 and 2x + y + 3z = 0. Determine whether N is a regular submanifold of R3. (6) Give the definition of a transverse map. Prove the transversality the- orem: if a smooth map f : M ! N is transverse to a regular submanifold S ⊂ N, then f −1 is a regular submanifold of M. (7) Give the definitions of an immersion and submersion. Which of the fol- lowing maps are immersions/submersions: • R2 ! R3 given by (x; y) 7! (x; y; 0) • R3 ! R2) given by (x; y; z) 7! (x; y) • R ! R2 given by x 7! (x2; x3). • R ! R2 given by x 7! (x2 − 1; x3 − x) • R ! R2 with image letter `b'. (8) Formulate the constant rank theorem and prove the immersion theorem: Suppose f : M ! N is an immersion at p 2 M. Then there are a coordinate chart (U; x1; :::; xm) about p and a coordinate chart about f(p) such that with respect to coordinate charts fjU is given by f(x1; :::; xm) = (x1; :::; xm; 0; :::; 0). Formulate and prove the submersion theorem. (9) Give the definition of an embedding. Show that the image of an embedding is a regular submanifold, and that the inclusion of a regular submanifold is an embedding. 4 DIFFERENTIAL GEOMETRY. PROBLEM LIST. 4. Tangent bundle (1) Give the definition of the tangent bundle TM of a manifold as a set. In- troduce a topology on TM and show that TM is a topological manifold. Introduce a smooth structure on TM. (2) Give the definition of a vector bundle. Show that for any smooth manifold M, the projection M × Rn ! M is a vector bundle (called trivial). Show that the projection TM ! M taking a pair (x; v) to x is also a vector bundle (called the tangent vector bundle). 5. Bump functions (1) Give the definition of a bump function on a manifold M at a point p 2 M. Define explicitly a bump function on a manifold at a point. (2) Prove that for any functions ρ1; :::; ρm on a manifold M, we have X supp( ρi) ⊂ [supp(ρi): (3) Let fKαg be a locally finite family of subsets of a topological space X. Show that every compact set K in X has a neighborhood that intersects only finitely many sets Kα. (4) Give the definition of a partition of unity. Prove that on a compact manifold with an open cover fUig there exists a partition of unity subordinate to fUig. 6. Vector fields (1) Give the definition of a smooth vector field. Show that a vector field X on a manifold M is smooth if and only if for every smooth function f on M, the function X(f) is also smooth.