Math 261A: Lie Groups, Fall 2008Problems, Set 21. (a) Show that the composition of two immersions is an immersion.(b) Show that an immersed submanifold N ⊆ M is always a closed submanifold of anopen submanifold, but not necessarily an open submanifold of a closed submanifold.2. Prove that if f : N → M is a smooth [analytic, holomorphic] map, then (df)pis surjec-tive if and only if there are open neighborhoods U of p and V of f(p), and an isomorphismψ : V × W → U, such that f ◦ ψ is the projection on V .In particular, deduce that the fibers of f meet a neighborhood of p in immersed closedsubmanifolds of that neighborhood.3. [corrected 9/30] Prove the implicit function theorem: a map (of sets) f : M → Nbetween manifolds is smooth [analytic, holomorphic] if and only if its graph H is an immersedclosed submanifold of M × N, and the tangent space to H at each point (x, f (x)) projectsisomorphically on the tangent space TxM.4. Prove that the curve y2= x3in R2is not an immersed submanifold. [This is a strongerstatement than the observation we made in class that the smooth bijection t 7→ (t2, t3) of Ronto this curve is not an immersion.]5. Let M be a complex holomorphic manifold, p a point of M, X a holomorphic vectorfield. Show that X has a complex integral curve γ defined on an open neighborhood U of 0in C, and unique on U if U is connected, which satisfies the usual defining equation but ina complex instead of a real variable t.Show that the restriction of γ to U ∩ R is a real integral curve of X, when M is regardedas a real analytic manifold. [This exercise is meant to clarify a point left vague in the lecture.]6. Let SL(2, C) act on the Riemann sphere P1(C) by fractional linear transformationsa bc dz = (az + b)/(cz + d). Determine explicitly the vector fields f(z)∂z corresponding tothe infinitesimal action of the basis elementsE =0 10 0, H =1 00 −1, F =0 01 0,of sl(2, C), and check that you have constructed a Lie algebra homomorphism by computingthe commutators of these vector fields.7. (a) Describe the map gl(n, R) = Lie(GL(n, R)) = Mn(R) → Vect(Rn) given by theinfinitesimal action of GLn(R).(b) Show that so(n, R) is equal to the subalgebra of gl(n, R) consisting of elements whoseinfinitesimal action is a vector field tangential to the unit sphere in Rn.8. (a) Let X be an analytic vector field on M all of whose integral curves are unbounded(i.e., they are defined on all of R). Show that there exists an analytic action of R = (R, +)on M such that X is the infinitesimal action of the generator ∂t of Lie(R).(b) More generally, prove the corresponding result for a family of n commuting vectorfields Xiand action of Rn.9. (a) Show that the matrix−a 00 −bbelongs to the identity component of GL(2, R)for all positive real numbers a, b.(b) Prove that if a 6= b, the above matrix is not in the image exp(gl(2, R)) of the expo-nential