Part 2 A Question list

The following list is an auto-generated copy from the past papers in the geometry aspect of the S.-T. Yau College Student Mathematics Contest. Please check with the original material at http://yau-contest.com/en/lists-jxxg.html.

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2010 - Individual

Problem 1: Let \(D^{*}=\left\{(x, y) \in \mathbb{R}^{2} \mid 0<x^{2}+y^{2}<1\right\}\) be the punctured unit disc in the Euclidean plane. Let \(g\) be the complete Riemannian metric on \(D^{*}\) with constant curvature -1. Find the distance under the metric between the points \(\left(e^{-2 \pi}, 0\right)\) and \(\left(-e^{-\pi}, 0\right)\).

Problem 2: Show that every closed hypersurface in \(\mathbb{R}^{n}\) has a point at which the second fundamental form is positive definite.

Problem 3: Prove that the real projective space \(\mathbb{R} P^{n}\) is orientable if and only if \(n\) is odd.

Problem 4: Suppose \(\pi: M_{1} \longrightarrow M_{2}\) is a \(C^{\infty}\) map of one connected differentiable manifold to another. And suppose for each \(p \in M_{1}\), the differential \(\pi_{*}: T_{p} M_{1} \longrightarrow T_{\pi(p)} M_{2}\) is a vector space isomorphism. (a). Show that if \(M_{1}\) is connected, then \(\pi\) is a covering space projection. (b). Given an example where \(M_{2}\) is compact but \(\pi: M_{1} \longrightarrow M_{2}\) is not a covering space (but has the \(\pi_{*}\) isomorphism property).

Problem 5: Let \(\Sigma_{g}\) be the closed orientable surface of genus \(g\). Show that if \(g>1\), then \(\Sigma_{g}\) is a covering space of \(\Sigma_{2}\).

Problem 6: Let \(M\) be a smooth 4-dimensional manifold. A symplectic form is a closed 2-form \(\omega\) on \(M\) such that \(\omega \wedge \omega\) is a nowhere vanishing 4-form. (a). Construct a symplectic form on \(\mathbb{R}^{4}\). (b). Show that there are no symplectic forms on \(S^{4}\).

2010 - Team

Problem 1: Let \(S^{n} \subset \mathbb{R}^{n+1}\) be the unit sphere, and \(\mathbb{R}^{n} \subset \mathbb{R}^{n+1}\) the equator \(n\) plane through the center of \(S^{n}\). Let \(N\) be the north pole of \(S^{n}\). Define a mapping \(\pi: S^{n} \backslash\{N\} \rightarrow \mathbb{R}^{n}\) called the stereographic projection that takes \(A \in S^{n} \backslash\{N\}\) into the intersection \(A^{\prime} \in \mathbb{R}^{n}\) of the equator \(n\) plane \(\mathbb{R}^{n}\) with the line which passes through \(A\) and \(N\). Prove that the stereographic projection is a conformal change, and derive the standard metric of \(S^{n}\) by the stereographic projection.

Problem 2: Let \(M\) be a (connected) Riemannian manifold of dimension 2. Let \(f\) be a smooth non-constant function on \(M\) such that \(f\) is bounded from above and \(\Delta f \geq 0\) everywhere on \(M\). Show that there does not exist any point \(p \in M\) such that \(f(p)=\sup \{f(x): x \in M\}\).

Problem 3: Let \(M\) be a compact smooth manifold of dimension \(d\). Prove that there exists some \(n \in \mathbb{Z}^{+}\) such that \(M\) can be regularly embedded in the Euclidean space \(\mathbb{R}^{n}\).

Problem 4: Show that any \(C^{\infty}\) function f on a compact smooth manifold \(M\) (without boundary) must have at least two critical points. When \(M\) is the 2-torus, show that \(f\) must have more than two critical points.

Problem 5: Construct a space \(X\) with \(H_{0}(X)=\mathbb{Z}, H_{1}(X)=\mathbb{Z}_{2} \times \mathbb{Z}_{3}, H_{2}(X)=\mathbb{Z}\), and all other homology groups of \(X\) vanishing.

Problem 6: (a). Define the degree \(\operatorname{deg} f\) of a \(C^{\infty} \operatorname{map} f: S^{2} \longrightarrow S^{2}\) and prove that \(\operatorname{deg} f\) as you present it is well-defined and independent of any choices you need to make in your definition. (b). Prove in detail that for each integer \(k\) (possibly negative), there is a \(C^{\infty}\) map \(f: S^{2} \longrightarrow S^{2}\) of degree \(k\).

2011 - Individual

Problem 1: Suppose \(M\) is a closed smooth \(n\)-manifold. (a) Does there always exist a smooth map \(f: M \rightarrow S^{n}\) from \(M\) into the \(n\)-sphere, such that \(f\) is essential (i.e. \(f\) is not homotopic to a constant map)? Justify your answer. (b) Same question, replacing \(S^{n}\) by the n-torus \(T^{n}\).

Problem 2: Suppose \((X, d)\) is a compact metric space and \(f: X \rightarrow X\) is a map so that \(d(f(x), f(y))=d(x, y)\) for all \(x, y\) in \(X\). Show that f is an onto map.

Problem 3: Let \(C_{1}, C_{2}\) be two linked circles in \(\mathbb{R}^{3}\). Show that \(C_{1}\) cannot be homotopic to a point in \(\mathbb{R}^{3} \backslash C_{2}\).

Problem 4: Let \(M=\mathbb{R}^{2} / \mathbb{Z}^{2}\) be the two dimensional torus, \(L\) the line \(3 x=7 y\) in \(\mathbb{R}^{2}\), and \(S=\pi(L) \subset M\) where \(\pi: \mathbb{R}^{2} \rightarrow M\) is the projection map. Find a differential form on \(M\) which represents the Poincaré dual of \(S\).

Problem 5: A regular curve \(C\) in \(\mathbb{R}^{3}\) is called a Bertrand Curve, if there exists a diffeomorphism \(f: C \rightarrow D\) from \(C\) onto a different regular curve \(D\) in \(\mathbb{R}^{3}\) such that \(N_{x} C=N_{f(x)} D\) for any \(x \in C\). Here \(N_{x} C\) denotes the principal normal line of the curve \(C\) passing through \(x\), and \(T_{x} C\) will denote the tangent line of \(C\) at \(x\). Prove that: (a) The distance \(|x-f(x)|\) is constant for \(x \in C\); and the angle made between the directions of the two tangent lines \(T_{x} C\) and \(T_{f(x)} D\) is also constant. (b) If the curvature \(k\) and torsion \(\tau\) of \(C\) are nowhere zero, then there must be constants \(\lambda\) and \(\mu\) such that \(\lambda k+\mu \tau=1\)

Problem 6: Let \(M\) be the closed surface generated by carrying a small circle with radius \(r\) around a closed curve \(C\) embedded in \(\mathbb{R}^{3}\) such that the center moves along \(C\) and the circle is in the normal plane to \(C\) at each point. Prove that \(\int_{M} H^{2} d \sigma \geq 2 \pi^{2}\) and the equality holds if and only if \(C\) is a circle with radius \(\sqrt{2} r\). Here \(H\) is the mean curvature of \(M\) and \(d \sigma\) is the area element of \(M\).

2011 - Team

Problem 1: Suppose K is a finite connected simplicial complex. True or false: (a) If \(\pi_{1}(K)\) is finite, then the universal cover of \(K\) is compact. (b) If the universal cover of K is compact then \(\pi_{1}(K)\) is finite.

Problem 2: Compute all homology groups of the the m-skeleton of an n-simplex, \(0 \leq m \leq n\).

Problem 3: Let \(M\) be an \(n\)-dimensional compact oriented Riemannian manifold with boundary and \(X\) a smooth vector field on \(M\). If \(\mathbf{n}\) is the inward unit normal vector of the boundary, show that \(\int_{M} \operatorname{div}(X) d V_{M}=\int_{\partial M} X \cdot \mathbf{n} d V_{\partial M}\).

Problem 4: Let \(\mathcal{F}^{k}(M)\) be the space of all \(C^{\infty} k\)-forms on a differentiable manifold \(M\). Suppose \(U\) and \(V\) are open subsets of \(M\). (a) Explain carefully how the usual exact sequence \(0 \longrightarrow \mathcal{F}(U \cup V) \longrightarrow \mathcal{F}(U) \oplus \mathcal{F} V) \longrightarrow \mathcal{F}(U \cap V) \longrightarrow 0\) arises. (b) Write down the “long exact sequence” in de Rham cohomology associated to the short exact sequence in part (a) and describe explicitly how the map \(H_{d e R}^{k}(U \cap V) \longrightarrow H_{d e R}^{k+1}(U \cup V)\) arises.

Problem 5: Let \(M\) be a Riemannian \(n\)-manifold. Show that the scalar curvature \(R(p)\) at \(p \in M\) is given by \(R(p)=\frac{1}{\operatorname{vol}\left(S^{n-1}\right)} \int_{S^{n-1}} \operatorname{Ric}_{p}(x) d S^{n-1}\) where \(\operatorname{Ric}_{p}(x)\) is the Ricci curvature in direction \(x \in S^{n-1} \subset T_{p} M\), \(\operatorname{vol}\left(S^{n-1}\right)\) is the volume of \(S^{n-1}\) and \(d S^{n-1}\) is the volume element of \(S^{n-1}\).

Problem 6: Prove the Schur’s Lemma: If on a Riemannian manifold of dimension at least three, the Ricci curvature depends only on the base point but not on the tangent direction, then the Ricci curvature must be constant everywhere, i.e., the manifold is Einstein.

2012 - Individual

Problem 1: Show that \(\pi_{3}\left(S^{2}\right) \neq 0\).

Problem 2: Let \(M\) be a smooth manifold of dimension \(n\), and \(X_{1}, \cdots, X_{k}\) be \(k\) everywhere linearly independent smooth vector fields on an open set \(U \subset M\) satisfying that \(\left[X_{i}, X_{j}\right]=0\) for \(1 \leq i, j \leq k\). Prove that for any point \(p \in U\) there is a coordinate chart \(\left(V, y^{i}\right)\) with \(p \in V \subseteq U\) and coordinates \(\left\{y^{1}, \cdots, y^{n}\right\}\) such that \(X_{i}=\frac{\partial}{\partial y^{i}}\) on \(V\) for each \(1 \leq i \leq k\).

Problem 3: Show that any self homeomorphism of \(\mathbb{C P}^{2}\) is orientation preserving.

Problem 4: Prove the following version of the isoperimetric inequality: Suppose \(C\) is a simple (that is, without self-intersection), smooth, closed curve in the Euclidean plane, with length \(L\). Show that the area enclosed by \(C\) is less than or equal to \(\frac{L^{2}}{4 \pi}\), and the equality occurs when and only when \(C\) is a round circle.

Problem 5: Let \(x: M \rightarrow \mathbb{R}^{3}\) be a closed surface in 3-dimensional Euclidean space. Its Gaussian curvature and mean curvature are denoted by \(K\) and \(H\) respectively. Prove that: \(\iint_{M} H d A+\iint_{M} p K d A=0, \quad \iint_{M} p H d A+\iint_{M} d A=0\) where \(p=\vec{x} \cdot \vec{n}\) is the support function of \(M, \vec{x}\) denotes the position vector of \(M, \vec{n}\) denotes the unit normal to \(M\), and \(d A\) is the area element of \(M\).

Problem 6: Write the structure equation of an orthonormal frame on a Riemannian manifold. Prove the following Riemannian metric \(g\) has constant sectional curvature \(c\) using the structure equation: \(g=\frac{\sum_{i=1}^{n}\left(d x^{i}\right)^{2}}{\left[1+\frac{c}{4} \sum_{i=1}^{n}\left(x^{i}\right)^{2}\right]^{2}}\) where \(\left(x^{1}, \ldots, x^{n}\right)\) is a local coordinate system.

2012 - Team

Problem 1: Prove that the real projective space \(\mathbb{R P}^{n}\) is a differentiable manifold of dimension \(n\).

Problem 2: Let \(M, N\) be \(n\)-dimensional smooth, compact, connected manifolds, and \(f: M \rightarrow N\) a smooth map with rank equals to \(n\) everywhere. Show that \(f\) is a covering map.

Problem 3: Given any Riemannian manifold \(\left(M^{n}, g\right)\), show that there exists a unique Riemannian connection on \(M^{n}\).

Problem 4: Let \(S^{n}\) be the unit sphere in \(\mathbb{R}^{n+1}\) and \(f: S^{n} \rightarrow S^{n}\) a continuous map. Assume that the degree of \(f\) is an odd integer. Show that there exists \(x_{0} \in S^{n}\) such that \(f\left(-x_{0}\right)=-f\left(x_{0}\right)\).

Problem 5: State and prove the Stokes theorem for oriented compact manifolds.

Problem 6: Let \(M\) be a surface in \(\mathbb{R}^{3}\). Let \(D\) be a simply-connected domain in \(M\) such that the boundary \(\partial D\) is compact and consists of a finite number of smooth curves. Prove the Gauss-Bonnet Formula: \(\int_{\partial D} k_{g} d s+\sum_{j}\left(\pi-\alpha_{j}\right)+\iint_{D} K d A=2 \pi\) where \(k_{g}\) is the geodesic curvature of the boundary curve. Each \(\alpha_{j}\) is the interior angle at a vertex of the boundary, \(K\) is the Gaussian curvature of \(M\), and the 2-form \(d A\) is the area element of \(M\).

2013 - Individual

Problem 1: Find the homology and fundamental group of the space \(X=S^{1} \times S^{1} /\{p, q\}\) obtained from the torus by identifying two distinct points \(p, q\) to one point.

Problem 2: Suppose \((X, d)\) is a compact metric space and \(f: X \rightarrow X\) is a map so that \(d(f(x), f(y))=d(x, y)\) for all \(x, y \in X\). Show that \(f\) is an onto map.

Problem 3: Let \(M^{2}\) ba a complete regular surface and \(K\) be the Gaussian curvature. Suppose \(\sigma:[0, \infty) \rightarrow M\) is a geodesic such that \(K(\sigma(t)) \leq f(t)\), where \(f\) is a differentiable function on \([0, \infty)\). Prove that any solution \(u(t)\) of the equation \(u^{\prime \prime}(t)+f(t) u(t)=0\) has a zero on \(\left[0, t_{0}\right]\), where \(\sigma\left(t_{0}\right)\) is the first conjugate point to \(\sigma(0)\) along \(\sigma\).

Problem 4: Let \(g_{1}, g_{2}\) be Riemannian metrics on a differentiable manifold \(M\), and denote by \(R_{1}\) and \(R_{2}\) their respective Riemannian curvature tensor. Suppose that \(R_{1}(X, Y, Y, X)=R_{2}(X, Y, Y, X)\) holds for any tangent vectors \(X, Y \in T_{p} M\). Show that \(R_{1}(X, Y, Z, W)=R_{2}(X, Y, Z, W)\) for any \(X, Y, Z, W \in T_{p} M\).

Problem 5: Let \(M^{n}\) be an even dimensional, orientable Riemannian manifold with positive sectional curvature. Let \(\sigma:[0, l] \rightarrow M\) be a closed geodesic, namely, \(\sigma\) is a geodesic with \(\sigma(0)=\sigma(l)\) and \(\sigma^{\prime}(0)=\sigma^{\prime}(l)\). Show that there exist an \(\epsilon>0\) and a smooth map \(F:[0, l] \times(-\epsilon, \epsilon) \rightarrow M\), such that \(F(t, 0)=\sigma(t)\), and for any fixed \(s \neq 0\) in \((-\epsilon, \epsilon), \sigma_{s}(t)=F(t, s)\) is a closed smooth curve with length less than that of \(\sigma\).

Problem 6: Let \(\left(M^{2}, d s^{2}\right)\) be a minimal surface in \(\mathbb{R}^{3}\), where \(d s^{2}\) is the restriction of the Euclidean metric. Assume that the Gaussian curvature \(K\) of \(\left(M^{2}, d s^{2}\right)\) is negative. Denote by \(\widetilde{K}\) the Gaussian curvature of the metric \(\widetilde{d s^{2}}=-K d s^{2}\). Show that \(\widetilde{K}=1\).

2013 - Team

Problem 1: Let \(X\) be the space \(\left\{(x, y, 0) \mid x^{2}+y^{2}=1\right\} \cup \left\{(x, 0, z) \mid x^{2}+z^{2}=1\right\}\). Find the fundamental group \(\pi_{1}\left(\mathbb{R}^{3} \backslash X\right)\).

Problem 2: Let \(M\) be a smooth connected manifold and \(f: M \rightarrow M\) be an injective smooth map such that \(f \circ f=f\). Show that the image set \(f(M)\) is a smooth submanifold in \(M\).

Problem 3: Let \(T^{2}=\left\{(z, w) \in \mathbb{C}^{2}\mid|z|=1,|w|=1\right\}\) be the torus. Define a map \(f: T^{2} \rightarrow T^{2}\) by \(f(z, w)=\left(z w^{3}, w\right)\). Prove that \(f\) is a diffeomorphism.

Problem 4: Prove: Any 3-dimensional Einstein manifold has constant curvature.

Problem 5: State and prove the Myers theorem for complete Riemannian manifolds.

Problem 6: Let \(C\) be a regular closed curve in \(\mathbb{R}^{3}\). Its torsion is \(\tau\). The integral \(\frac{1}{2 s} \int_{C} \tau d s\) is called the total torsion of \(C\), where \(s\) is the arc length parameter. Prove: Given a smooth surface \(M\) in \(\mathbb{R}^{3}\), if for any regular closed curve \(C\) on \(M\), the total torsion of \(C\) is always an integer, then \(M\) is a part of a sphere or a plane.

2014 - Individual

Problem 1: Let \(X\) be the quotient space of \(S^{2}\) under the identifications \(x \sim-x\) for \(x\) in the equator \(S^{1}\). Compute the homology groups \(H_{n}(X)\). Do the same for \(S^{3}\) with antipodal points of the equator \(S^{2} \subset S^{3}\) identified.

Problem 2: Let \(M \rightarrow \mathbb{R}^{3}\) be a graph defined by \(z=f(u, v)\) where \(\{u, v, z\}\) is a Descartes coordinate system in \(\mathbb{R}^{3}\). Suppose that \(M\) is a minimal surface. Prove that: (a) The Gauss curvature \(K\) of \(M\) can be expressed as \(K=\Delta \log \left(1+\frac{1}{W}\right), \quad W:=\sqrt{1+\left(\frac{\partial f}{\partial u}\right)^{2}+\left(\frac{\partial f}{\partial v}\right)^{2}}\) where \(\Delta\) denotes the Laplacian with respect to the induce metric on \(M\) (i.e., the first fundamental form of \(M\)). (b) If \(f\) is defined on the whole \(u v\)-plane, then \(f\) is a linear function (Bernstein theorem).

Problem 3: Let \(M=\mathbb{R}^{2} / \mathbb{Z}^{2}\) be the two dimensional torus, \(L\) the line \(3 x=7 y\) in \(\mathbb{R}^{2}\), and \(S=\pi(L) \subset M\) where \(\pi: \mathbb{R}^{2} \rightarrow M\) is the projection map. Find a differential form on \(M\) which represents the Poincaré dual of \(S\).

Problem 4: Let \(p:(\tilde{M}, \tilde{g}) \rightarrow(M, g)\) be a Riemannian submersion. This is a submersion \(p: \tilde{M} \rightarrow M\) such that for each \(x \in \tilde{M}, D p: \operatorname{ker}^{\perp}(D p) \rightarrow T_{p(x)}(M)\) is a linear isometry. (a) Show that \(p\) shortens distances. (b) If \((\tilde{M}, \tilde{g})\) is complete, so is \((M, g)\). (c) Show by example that if \((M, g)\) is complete, \((\tilde{M}, \tilde{g})\) may not be complete.

Problem 5: Let \(\Psi: M \rightarrow \mathbb{R}^{3}\) be an isometric immersion of a compact surface \(M\) into \(\mathbb{R}^{3}\). Prove that \(\int_{M} H^{2} d \sigma \geq 4 \pi\), where \(H\) is the mean curvature of \(M\) and \(d \sigma\) is the area element of \(M\).

Problem 6: The unit tangent bundle of \(S^{2}\) is the subset \(T^{1}\left(S^{2}\right)=\left\{(p, v) \in \mathbb{R}^{3} \mid\|p\|=1,(p, v)=0 \text { and }\|v\|=1\right\}\). Show that it is a smooth submanifold of the tangent bundle \(T\left(S^{2}\right)\) of \(S^{2}\) and \(T^{1}\left(S^{2}\right)\) is diffeomorphic to \(\mathbb{R} P^{3}\).

2014 - Team

Problem 1: Compute the fundamental and homology groups of the wedge sum of a circle \(S^{1}\) and a torus \(T=S^{1} \times S^{1}\).

Problem 2: Given a properly discontinuous action \(F: G \times M \rightarrow M\) on a smooth manifold \(M\), show that \(M / G\) is orientable if and only if \(M\) is orientable and \(F(g, \cdot)\) preserves the orientation of \(M\). Use this statement to show that the Möbius band is not orientable and that \(\mathbb{R} P^{n}\) is orientable if and only if \(n\) is odd.

Problem 3: (a) Consider the space \(Y\) obtained from \(S^{2} \times[0,1]\) by identifying \((x, 0)\) with \((-x, 0)\) and also identifying \((x, 1)\) with \((-x, 1)\), for all \(x \in S^{2}\). Show that \(Y\) is homeomorphic to the connected sum \(\mathbb{R} P^{3} \# \mathbb{R} P^{3}\). (b) Show that \(S^{2} \times S^{1}\) is a double cover of the connected sum \(\mathbb{R} P^{3} \# \mathbb{R} P^{3}\).

Problem 4: Prove that a bi-invariant metric on a Lie group \(G\) has nonnegative sectional curvature.

Problem 5: Let \(M\) be the upper half-plane \(\mathbb{R}_{+}^{2}\) with the metric \(d s^{2}=\frac{d x^{2}+d y^{2}}{y^{k}}\). For which values of \(k\) is \(M\) complete?

Problem 6: Given any nonorientable manifold \(M\) show the existence of a smooth orientable manifold \(\bar{M}\) which is a double covering of \(M\). Find \(\bar{M}\) when \(M\) is \(\mathbb{R} P^{2}\) or the Möbius band.

2015 - Individual

Problem 1: Let \(n, m\) be positive integers. Show that the product of spheres \(S^{n} \times S^{m}\) has trivial tangent bundle if and only if \(n\) or \(m\) is odd.

Problem 2: Show that there does not exist a compact three-dimensional manifold \(M\) whose boundary is the real projective space \(\mathbb{R P}^{2}\).

Problem 3: Let \(M^{n}\) be a smooth manifold without boundary and \(X\) a smooth vector field on \(M\). If \(X\) does not vanish at \(p \in M\), show that there exists a local coordinate chart \(\left(U ; x_{1}, \ldots, x_{n}\right)\) centered at \(p\) such that in \(U\) the vector field \(X\) takes the form \(X=\frac{\partial}{\partial x_{i}}\).

Problem 4: Let \(M \rightarrow \mathbb{R}^{3}\) be a compact simply-connected closed surface. Prove that if \(M\) has constant mean curvature, then \(M\) is a standard sphere.

Problem 5: Let \(M\) be an \(n\)-dimensional compact Riemannian manifold with diameter \(\pi / c\) and Ricci curvature \(\geq(n-1) c^{2}>0\). Show that \(M\) is isometric to the standard \(n\)-sphere in \(\mathbb{R}^{n+1}\) with radius \(1 / c\).

Problem 6: Suppose \((M, g)\) is a Riemannian manifold and \(p \in M\). Show that the second-order Taylor series of \(g\) in normal coordinates centered at \(p\) is \(g_{i j}(x)=\delta_{i j}-\frac{1}{3} \sum_{k, l} R_{i k l j} x_{k} x_{l}+O\left(|x|^{3}\right)\).

2015 - Team

Problem 1: Let \(S O(3)\) be the set of all \(3 \times 3\) real matrices \(A\) with determinant 1 and satisfying \({ }^{t} A A=I\), where \(I\) is the identity matrix and \({ }^{t} A\) is the transpose of \(A\). Show that \(S O(3)\) is a smooth manifold, and find its fundamental group. You need to prove your claims.

Problem 2: Let \(X\) be a topological space. The suspension \(S(X)\) of \(X\) is the space obtained from \(X \times[0,1]\) by contracting \(X \times\{0\}\) to a point and contracting \(X \times\{1\}\) to another point. Describe the relation between the homology groups of \(X\) and \(S(X)\).

Problem 3: Let \(F: M \rightarrow N\) be a smooth map between two manifolds. Let \(X_{1}, X_{2}\) be smooth vector fields on \(M\) and let \(Y_{1}, Y_{2}\) be smooth vector fields on \(N\). Prove that if \(Y_{1}=F_{*} X_{1}\) and \(Y_{2}=F_{*} X_{2}\), then \(F_{*}\left[X_{1}, X_{2}\right]=\left[Y_{1}, Y_{2}\right]\), where [, ] is the Lie bracket.

Problem 4: Let \(M_{1}\) and \(M_{2}\) be two compact convex closed surfaces in \(\mathbb{R}^{3}\), and \(f: M_{1} \rightarrow M_{2}\) a diffeomerphism such that \(M_{1}\) and \(M_{2}\) have the same inner normal vectors and Gauss curvatures at the corresponding points. Prove that \(f\) is a translation.

Problem 5: Prove the second Bianchi identity: \(R_{i j k l, h}+R_{i j l h, k}+R_{i j h k, l}=0\).

Problem 6: Let \(M_{1}, M_{2}\) be two complete \(n\)-dimensional Riemannian manifolds and \(\gamma_{i}:[0, a] \rightarrow M_{i}\) are two arc length parametrized geodesics. Let \(\rho_{i}\) be the distance function to \(\gamma_{i}(0)\) on \(M_{i}\). Assume that \(\gamma_{i}(a)\) is within the cut locus of \(\gamma_{i}(0)\) and for any \(0 \leq t \leq a\) we have the inequality of sectional curvatures \(K_{1}\left(X_{1}, \frac{\partial}{\partial \gamma_{1}}\right) \geq K_{2}\left(X_{2}, \frac{\partial}{\partial \gamma_{2}}\right)\) where \(X_{i} \in T_{\gamma_{i}(t)} M_{i}\) is any unit vector orthogonal to the tangent \(\frac{\partial}{\partial \gamma_{i}}\). Then \(\operatorname{Hess}\left(\rho_{1}\right)\left(\widetilde{X}_{1}, \widetilde{X}_{1}\right) \leq \operatorname{Hess}\left(\rho_{2}\right)\left(\widetilde{X}_{2}, \widetilde{X}_{2}\right)\) where \(\widetilde{X}_{i} \in T_{\gamma_{i}(a)} M_{i}\) is any unit vector orthogonal to the tangent \(\frac{\partial}{\partial \gamma_{i}}(a)\).

2016 - Individual

Problem 1: Let \(M\) be a compact odd-dimensional manifold with boundary \(\partial M\). Show that the Euler characteristics of \(M\) and \(\partial M\) are related by: \(\chi(M)=\frac{1}{2} \chi(\partial M)\).

Problem 2: Compute the de Rham cohomology of a punctured two-dimensional torus \(T^{2}-\{p\}\), where \(p \in T^{2}\). If \(T^{2}=\mathbb{R}^{2} / \mathbb{Z}^{2}\) with coordinates \((x, y) \in \mathbb{R}^{2}\), then is the volume form \(\omega=d x \wedge d y\) exact?

Problem 3: Let \(M^{n} \rightarrow \mathbb{R}^{n+1}\) be a closed oriented hypersurface. The \(r\)-th mean curvature of \(M^{n}\) is defined by \(H_{r}:=\frac{1}{\binom{n}{r}} \sum_{i_{1}<i_{2} \cdots<i_{r}} \lambda_{i_{1}} \lambda_{i_{2}} \cdots \lambda_{i_{r}}, \quad(1 \leq r \leq n)\) where \(\lambda_{i}(i=1, \cdots, n)\) are principal curvatures of \(M^{n}\). Prove that if all of \(\lambda_{i}\) are positive and \(H_{r}=\) constant for a certain \(r\), then \(M^{n}\) is a hypersphere in \(\mathbb{R}^{n+1}\).

Problem 4: State and prove the cut-off function lemma on a differentiable manifold.

Problem 5: Let \(M\) be a compact Riemannian manifold without boundary. Show that if \(M\) has positive Ricci curvature, then \(H^{1}(M, \mathbb{R})=0\).

Problem 6: Let \(M\) be an orientable, closed and embedded minimal hypersurface in \(S^{n+1}\). Denote by \(\lambda_{1}\) the first eigenvalue for the Laplace-Beltrami operator on \(M\). Prove that \(\lambda_{1} \geq n / 2\).

2016 - Team

Problem 1: Show that \(\mathbb{C P}^{2 n}\) does not cover any manifold except itself.

Problem 2: Let \(X\) be a topological space and \(p \in X\). The reduced suspension \(\Sigma X\) of \(X\) is the space obtained from \(X \times[0,1]\) by contracting \((X \times\{0,1\}) \cup(\{p\} \times[0,1])\) to a point. Describe the relation between the homology groups of \(X\) and \(\Sigma X\).

Problem 3: State and prove the Frobenius Theorem on a differentiable manifold.

Problem 4: Show that all geodesics on the sphere \(S^{n}\) are precisely the great circles.

Problem 5: Let \(M\) be an n-dimensional Riemannian manifold. Denote by \(R\) and \(K_{M}\) the curvature tensor and sectional curvature of \(M\). If \(a \leq K_{M} \leq b\) at a point \(x \in M\), then, at this point, \(R\left(e_{1}, e_{2}, e_{3}, e_{4}\right) \leq \frac{2}{3}(b-a)\) for all orthonormal four-frames \(\left\{e_{1}, e_{2}, e_{3}, e_{4}\right\} \subset T_{x} M\).

Problem 6: Let \(M\) be a closed minimal hypersurface with constant scalar curvature in \(S^{n+1}\). Denote by \(S\) the squared length of the second fundamental form of \(M\). Show that \(S=0\), or \(S \geq n\).

2017 - Individual

Problem 1: Let \(M\) be a smooth, compact, oriented \(n\)-dimensional manifold. Suppose that the Euler characteristic of \(M\) is zero. Show that \(M\) admits a nowhere vanishing vector field.

Problem 2: Let \(S^{2} \stackrel{q_{1}}{\sim} S^{2} \vee S^{2} \stackrel{q_{2}}{\sim} S^{2}\) be the maps that crush out one of the two summands. Let \(f: S^{2} \rightarrow S^{2} \vee S^{2}\) be a map such that \(q_{i} \circ f: S^{2} \rightarrow S^{2}\) is a map of degree \(d_{i}\). Compute the integral homology groups of \(\left(S^{2} \vee S^{2}\right) \cup_{f} D^{3}\). Here \(D^{3}\) is the unit solid ball with boundary \(S^{2}\).

Problem 3: Let \(X\) and \(Y\) be smooth vector fields on a smooth manifold. Prove that the Lie derivative satisfies the identity \(L_{X} Y=[X, Y]\).

Problem 4: State and prove the Liouville formula for the geodesic curvature \(\kappa_{g}\) along a regular curve on a smooth surface in \(\mathbb{R}^{3}\).

Problem 5: On a Riemannian manifold, let \(F\) be the set of smooth functions \(f\) on \(M\) with \(|\operatorname{grad} f| \leq 1\). For any \(x, y\) in the manifold, show that \(d(x, y)=\sup \{|f(x)-f(y)|: f \in F\}\).

Problem 6: Let \(M\) be an \(n\)-dimensional oriented closed minimal submanifold in an \((n+p)\) dimensional unit sphere \(S^{n+p}\). Denote by \(K_{M}\) the sectional curvature of \(M\). Prove that if \(K_{M}>\frac{p-1}{2 p-1}\), then \(M\) is the great sphere \(S^{n}\).

2017 - Team

Problem 1: Consider the space \(X=M_{1} \cup M_{2}\), where \(M_{1}\) and \(M_{2}\) are Möbius bands and \(M_{1} \cap M_{2}=\partial M_{1}=\partial M_{2}\). Here a Möbius band is the quotient space \(\left([-1,1] \times[-1,1]\right) /\left((1, y) \sim(-1,-y)\right)\). Determine the fundamental group of \(X\).

Problem 2: If \(f: X \rightarrow X\) is a self-map, then the “mapping torus of \(f\)” is the quotient \(T_{f}:=(X \times[0,1]) /(x, 0) \sim(f(x), 1), \quad \forall x \in X\). For \(n \in \mathbb{Z}\), let \(f_{n}\) be a degree \(n\) map on \(S^{3}\). Compute the integral homology groups of \(T_{f_{n}}\).

Problem 3: Let \(C\) be a regular curve on a smooth surface \(S\) in \(R^{3}\). Denote by \(I=E d u^{2}+2 F d u d v+G d v^{2}\) and \(I I=L d u^{2}+2 M d u d v+N d v^{2}\) the first and second fundamental forms of \(S\), respectively. Assume that the equation of \(C\) is given by \(u=u(s), v=v(s)\), where \(s\) is the arc-length parameter of \(C\). Show that geodesic torsion along the curve \(C\) satisfies \(\tau_{g}=\frac{1}{\sqrt{E G-F^{2}}}\left|\begin{array}{ccc}\left(\frac{d v}{d s}\right)^{2} & -\frac{d u}{d s} \frac{d v}{d s} & \left(\frac{d u}{d s}\right)^{2} \\ E & F & G \\ L & M & N \\ \end{array}\right|\).

Problem 4: Let \(\left\{e_{i}\right\}_{i=1, \ldots, n}\) be a basis of a vector space \(V\). Denote by \(\left\{\omega^{i}\right\}_{i=1, \ldots, n}\) the dual basis of \(\left\{e_{i}\right\}_{i=1, \ldots, n}\). Show that the set \(\left\{\omega^{i_{1}} \wedge \cdots \wedge \omega^{i_{r}} \mid 1 \leq i_{1}<i_{2}<\cdots<i_{r} \leq n\right\}\) is a basis of \(\bigwedge^{r} V^{*}\), where \(r\) is a positive integer and \(r \leq n\).

Problem 5: Let \(M \rightarrow R^{n+1}\) be a compact closed hypersurface in the \((n+1)\)-dimensional Euclidean space \(R^{n+1}\). Prove that \(M\) is a hypersphere if \(M\) has constant scalar curvature and nonnegative Ricci curvature.

Problem 6: On a Riemannian manifold, if \(f\) is a smooth function such that \(|\operatorname{grad} f|=1\). Show that the integral curves of \(\operatorname{grad} f\) are geodesics.

2018 - Individual

Problem 1: Let \(M\) and \(N\) be smooth, connected, orientable \(n\)-manifolds for \(n \geq 3\), and let \(M \# N\) denote their connect sum. (a) Compute the fundamental group of \(M \# N\) in terms of that of \(M\) and of \(N\) (you may assume that the basepoint is on the boundary sphere along which we glue \(M\) and \(N\)). (b) Compute the homology groups of \(M \# N\). (c) For part (a), what changes if \(n=2\)? Use this to describe the fundamental groups of orientable surfaces.

Problem 2: Determine all of the possible degrees of maps \(S^{2} \rightarrow S^{1} \times S^{1}\).

Problem 3: Classify all vector bundles over the circle \(S^{1}\) up to isomorphism.

Problem 4: Suppose \(C\) is a regular curve in the unit sphere \(S^{2}\). For any point \(W \in S^{2}\), there exists the only oriented great circle \(S_{W}\) (determined by the right hand rule) in \(S^{2}\) such that \(W\) is the pole of \(S_{W}\). Denote by \(n(W)\) the number of points at which the oriented great circle \(S_{W}\) and \(C\) intersect. Prove the Crofton formula \(\iint_{S^{2}} n(W) d W=4 L\) where \(d W\) and \(L\) is the area element of \(S^{2}\) and the length of \(C\), respectively.

Problem 5: Let \(M\) be an \(n\)-dimensional closed submanifold in the Euclidean space \(\mathbb{R}^{n+p}\). Prove the following inequality \(\int_{M} H^{n} d V \geq \operatorname{vol}\left(S^{n}\right)\) where \(H\) and \(d V\) is the mean curvature (i.e., norm of the mean curvature vector) and the volume element of \(M\), and \(S^{n}\) is the standard unit sphere of dimension \(n\).

Problem 6: Let \(M\) be an even dimensional compact and oriented Riemannian manifold with positive sectional curvature. Show that \(M\) is simply connected.

2018 - Team

Problem 1: Let \(X\) be \(\left(S^{2} \times S^{2}\right) \cup_{S^{2}} D^{3}\), where we attach the 3-disk via the map \(S^{2} \rightarrow S^{2} \vee S^{2}\) which crushes a great circle connecting the north and south poles. Compute the homology groups of \(X\).

Problem 2: (a) Let \(A\) be a single circle in \(\mathbb{R}^{3}\). Compute the fundamental group \(\pi_{1}\left(\mathbb{R}^{3}-A\right)\). (b) Let \(A\) and \(B\) be disjoint circles in \(\mathbb{R}^{3}\), supported in the upper and lower half space, respectively. Compute \(\pi_{1}\left(\mathbb{R}^{3}-(A \cup B)\right)\).

Problem 3: Consider the differential 1-form \(\omega=x d y-y d x+d z\) in \(\mathbb{R}^{3}\) with coordinates \((x, y, z)\). Prove that \(f \omega\) is not closed for any nowhere zero function \(f: \mathbb{R}^{3} \rightarrow \mathbb{R}\).

Problem 4: Show that \(Q^{n}:=\left\{\left(x^{1}, \cdots, x^{n+1}\right) \in \mathbb{R}^{n+1} ; \sum_{i=1}^{n+1}\left(x^{i}\right)^{4}=1\right\}\) is a differentiable manifold.

Problem 5: Let \(M\) be a closed surface in \(\mathbb{R}^{3}\). Prove that \(\int_{M}|K| d \sigma \geq 4 \pi(1+g)\) where \(K, g\) and \(d \sigma\) is the Gaussian curvature, the genus and the area element of \(M\), respectively.

Problem 6: Let \(M\) be an \(n\)-dimensional compact and simply connnected Riemannian manifold. If the sectional curvature \(K_{M}\) of \(M\) satisfies \(\frac{1}{4}<K_{M} \leq 1\) then \(M\) is homeomorphic to \(S^{n}\).

2019 - Individual

Problem 1: Let \(\operatorname{Conf}_{n}\) be the following submanifold of \(\mathbb{C}^{n}\): \(\operatorname{Conf}_{n}=\left\{\left(z_{1}, z_{2}, \cdots, z_{n}\right) \in \mathbb{C}^{n} \mid z_{i} \neq z_{j} \text { for any } i \neq j\right\}\). For every pair \((i, j)\) with \(i \neq j\), we define the complex valued 1-form \(\omega_{i j}:=\frac{d z_{i}-d z_{j}}{z_{i}-z_{j}}\). (a) Show that for any \(i \neq j, \omega_{i j}\) represents a non-zero de Rham cohomology class in \(H^{1}\left(\operatorname{Conf}_{n}, \mathbb{C}\right)\). (b) Show that for any pair-wise distinct indices \(i, j, k\), \(\omega_{i j} \wedge \omega_{j k}+\omega_{j k} \wedge \omega_{k i}+\omega_{k i} \wedge \omega_{i j}=0\).

Problem 2: Let \(M\) be a compact oriented manifold of (real) dimension 4. Consider the following symmetric bilinear form on \(H^{2}(M)\) \(H^{2}(M) \times H^{2}(M) \rightarrow \mathbb{R}, \quad([\alpha],[\beta]) \mapsto \int_{M} \alpha \wedge \beta\). Let \(\tau(M)\) be the signature of this bilinear form, i.e. the number of positive eigenvalues minus the number of negative eigenvalues. Compute \(\tau(M)\) for \(M=S^{4}, \mathbb{C P}^{2}\) and \(S^{2} \times S^{2}\).

Problem 3: Let \(X=\mathbb{R}^{4} / \sim\), where \(\begin{aligned} & \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \sim\left(x_{1}, x_{2}+1, x_{3}, x_{4}\right) \\ & \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \sim\left(x_{1}, x_{2}, x_{3}, x_{4}+1\right) \\ & \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \sim\left(x_{1}+1, x_{2}, x_{3}, x_{4}\right) \\ & \left(x_{1}, x_{2}, x_{3}, x_{4}\right) \sim\left(x_{1}, x_{2}+x_{4}, x_{3}+1, x_{4}\right) \end{aligned}\) Compute \(H_{1}(X, \mathbb{Z})\).

Problem 4: Let \(E\) be a vector bundle over a smooth manifold \(M\). Let \(\nabla^{E}\) be a connection \(E\) and \(R^{E} \in \Omega^{2}(M, \operatorname{End}(E))\) be its curvature tensor. For any polynomial \(f(x)=a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{n} x^{n}\), we denote \(f\left(R^{E}\right)=a_{0}+a_{1} R^{E}+a_{2}\left(R^{E}\right)^{2} \cdots+a_{n}\left(R^{E}\right)^{n} \in \Omega^{*}(M, \operatorname{End}(E))\). Here \(\left(R^{E}\right)^{k} \in \Omega^{2 k}(M, \operatorname{End}(E))\) is the \(k\)-th wedge product on forms combined with matrix multiplications on \(\operatorname{End}(E)\). (a) Show that the differential form \(\operatorname{tr}\left[f\left(R^{E}\right)\right] \in \Omega^{*}(M)\) is closed \(d \operatorname{tr}\left[f\left(R^{E}\right)\right]=0\). Here tr is the trace on \(\operatorname{End}(E)\). (b) Let \(\nabla^{E}, \widetilde{\nabla}^{E}\) be two connections on \(E\) and \(R^{E}, \widetilde{R}^{E}\) be the corresponding curvature tensors. Show that there exists a differential form \(\omega \in \Omega^{*}(M)\) such that \(\operatorname{tr}\left[f\left(R^{E}\right)\right]-\operatorname{tr}\left[f\left(\tilde{R}^{E}\right)\right]=d \omega\).

Problem 5: (a) Let \(u\) be a smooth function over a Riemannian manifold \((M, g)\). Prove the following Bochner’s formula \(\frac{1}{2} \Delta|\nabla u|^{2}=|\nabla \nabla u|^{2}+\operatorname{Ric}(\nabla u, \nabla u)+g(\nabla \Delta u, \nabla u)\) where \(\Delta\) is the Laplacian and \(|\bullet|^{2}=g(\bullet, \bullet)\). (b) Let \(\left(S^{2}, g\right)\) be the standard unit sphere and \(E\) be a constant. Show that the only smooth positive solution to \(\Delta \ln f+E f^{2}=1\) is \(f=\frac{1}{A+\phi}\) where \(A\) is a constant and \(\phi\) is some first eigenfunction of \(S^{2}\).

2019 - Team

Problem 1: Is \(T S^{2}\) diffeomorphic to \(S^{2} \times \mathbb{R}^{2}\)? Verify your answer. Here \(T S^{2}\) is the total space of the tangent bundle of \(S^{2}\).

Problem 2: Solve the problem which Russell Crowe assigns to his students in the movie “A beautiful mind” (2001): \(V=\left\{F: \mathbb{R}^{3} \backslash X \rightarrow \mathbb{R}^{3} \text { s.t. } \nabla \times F=0\right\}\) \(W=\{F=\nabla g\}\) \(\operatorname{dim}(V / W)=?\) First give the general answer for any closed \(X \subset \mathbb{R}^{3}\), and then specialize it to (a) \(X=\{x=y=z=0\}\), (b) \(X=\{x=y=0\}\) and (c) \(X=\{x=0\}\).

Problem 3: Let \(T^{2}=S^{1} \times S^{1}\) be the 2-torus with the standard orientation, and let \(F: T^{2} \rightarrow T^{2}\) be a smooth map of degree 1 such that \(F \circ F=\mathrm{Id}\) and \(F\) has no fixed points. Prove that the induced map \(F^{*}: H^{1}\left(T^{2}\right) \rightarrow H^{1}\left(T^{2}\right)\) is the identity.

Problem 4: Let \(U(n)\) be the group of \(n \times n\) unitary matrices, and \(O(n)\) be the group of \(n \times n\) orthogonal matrices. Let \(S U(n)=\{A \in U(n) \mid \operatorname{det} A=1\}\) be the special unitary group and \(S O(n)=\{A \in O(n) \mid \operatorname{det} A=1\}\) be the special orthogonal group. All \(U(n), S U(n), O(n), S O(n)\) are Lie groups with natural manifold structures. (a) Compute the dimensions of \(S U(n)\) and \(S O(n)\). (b) Compute the fundamental groups of \(S U(n)\) and \(S O(n)(n \geqslant 2)\).

Problem 5: Let \((M, g)\) be a compact Riemannian manifold and \(R\) be its Riemannian curvature tensor. \((M, g)\) will be called weakly negative if for any point \(p \in M\) and for any nonzero vector field \(X \in T_{p} M\), there exists a nonzero vector field \(Y \in T_{p} M\) such that \(R(X, Y, Y, X)<0\). (a) Let \(X\) be a Killing vector field and \(f=\frac{1}{2} g(X, X)=\frac{1}{2}|X|^{2}\). Show that for any vector field \(V\) \((\text { Hess } f)(V, V)=g\left(\nabla_{V} X, \nabla_{V} X\right)-R(V, X, X, V)\) Here the Hessian of \(f\) is \((\operatorname{Hess} f)(Y, Z):=g\left(\nabla_{Y} \operatorname{grad}(f), Z\right)\) for any vector fields \(Y, Z\), where \(\operatorname{grad}(f)\) is the gradient vector of \(f\). (b) Prove that if \((M, g)\) is weakly negative, then there are no nontrivial Killing vector fields.

2020 - Individual

Problem 1: Let \(S^{n}\) be the unit sphere in \(\mathbb{R}^{n+1}\). (a) Find a 6-form \(\alpha\) on \(\mathbb{R}^{7} \backslash\{0\}\) such that \(d \alpha=0, \quad \text { and } \quad \int_{S^{n}} \alpha=1\). (b) For any smooth map \(f: S^{11} \rightarrow S^{6}\), show that there exists a 5-form \(\varphi\) on \(S^{11}\) such that \(f^{*} \alpha=d \varphi\). (c) Let \(H(f)=\int_{S^{11}} \varphi \wedge d \varphi\). Show that \(H(f)\) is independent of the choice of \(\varphi\) satisfying (1). (d) Show that \(H(f)\) is an even integer, for any smooth map \(f: S^{11} \rightarrow S^{6}\).

Problem 2: For any \(h \in C^{\infty}\left(\mathbb{R}^{2}\right)\) and \(h>0\) on \(\mathbb{R}^{2}\), define the Ricci curvature \(\operatorname{Ric}(h)\) associated with \(h\) by \(\operatorname{Ric}(h)=\frac{1}{h}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^{2}}\right) \log h\) where \((x, y)\) are the standard Cartesian coordinates in \(\mathbb{R}^{2}\). Either construct a positive smooth function \(h_{1}\) such that \(\operatorname{Ric}\left(h_{1}\right)=1\), or show that no such function \(h_{1}\) exists.

Problem 3: Let \(M\) be an \(n\)-dimensional Riemannian manifold, and \(p \in M\). Let \(\left\{e_{1}, \ldots, e_{n}\right\}\) be an orthonormal basis of the tangent space \(T_{p} M\), and let \(\left\{x^{1}, \ldots, x^{n}\right\}\) be a coordinate system of \(M\) centered at \(p\) such that \(\exp _{p}^{-1}(q)=\sum_{j=1}^{n} x^{j}(q) e_{j}\) where \(\exp _{p}\) denotes the exponential map. Let \(\gamma(t)=\exp _{p}\left(t e_{1}\right), 0 \leq t \leq \delta\), where \(\delta\) is a positive constant less than 1. (a) For \(2 \leq \alpha \leq n\), which one of the following, \(\left.t \frac{\partial}{\partial x^{\alpha}}\right|_{\gamma(t)} \quad \text { or }\quad \left.\frac{\partial}{\partial x^{\alpha}}\right|_{\gamma(t)}\) is a Jacobi field along \(\gamma(t)\)? Prove your assertion. (b) Denote \(g_{i j}=\left\langle\frac{\partial}{\partial x^{i}}, \frac{\partial}{\partial x^{j}}\right\rangle, \quad 1 \leq i, j \leq n\) Compute \(\frac{\partial^{2} g_{22}}{\partial x^{1} \partial x^{1}} \quad \text { at the point } p\) (c) Show that \(\max _{0 \leq t \leq \delta}\left|\frac{\partial g_{22}}{\partial x^{1}}(\gamma(t))\right| \leq C \delta A\) where \(C>0\) is a constant depending only on \(n\), and \(A\) is the \(C^{0}\)-bound of the curvature tensor of \(M\) along \(\gamma(t)\), for \(0 \leq t \leq \delta\).

Problem 4: Let \(\mathrm{SO}(n)\) be the set of \(n \times n\) orthogonal real matrices with determinant equal to 1. Endow \(\mathrm{SO}(n)\) with the relative topology as a subspace of Euclidean space \(\mathbb{R}^{n^{2}}\). (a) Show that \(\mathrm{SO}(n)\) is compact. (b) Is \(\mathrm{SO}(3)\) homeomorphic to the real projective space \(\mathbb{R P}^{3}\)? Prove your assertion. (c) Compute the fundamental group of \(\mathrm{SO}(2020)\).

Problem 5: Let \(X\) be a topological space and \(\pi: \mathbb{R}^{2} \rightarrow X\) a covering map. Let \(K\) be a compact subset of \(X\) and \(B\) the closed unit ball centered at the origin in \(\mathbb{R}^{2}\). (a) Suppose \(\pi: \mathbb{R}^{2} \backslash B \rightarrow X \backslash K\) is a homeomorphism. Show that \(\pi: \mathbb{R}^{2} \rightarrow X\) is a homeomorphism. (b) Suppose \(\mathbb{R}^{2} \backslash B\) is homeomorphic to \(X \backslash K\), where the homeomorphism may not be \(\pi\). Is \(X\) necessarily homeomorphic to \(\mathbb{R}^{2}\)? Prove your assertion.

Problem 6: Let \(F_{n}\) be the free group of rank \(n\), (a) Give an example of a finite connected graph such that its fundamental group is \(F_{2}\). (b) Does \(F_{2}\) contain a proper subgroup isomorphic to \(F_{2}\)? (c) Does \(F_{2}\) contain a proper finite index subgroup isomorphic to \(F_{2}\)?

2021 - Individual

Problem 1: (a) Show that \(\mathbf{P}^{\mathbf{2 n}}\) can not be the boundary of a compact manifold. (b) Show that \(\mathbf{P}^{\mathbf{3}}\) is the boundary of some compact manifold.

Problem 2: Suppose \(M\) is a noncompact, complete \(n\)-dimensional manifold, and suppose there is an open subset \(U \subset M\) and an open set \(V \subset \mathbf{R}^{\mathbf{n}}\) such that \(M \backslash U\) is isomorphic to \(\mathbf{R}^{\mathbf{n}} \backslash V\). If Ric \(M \geq 0\), show that \(M\) is isometric to \(\mathbf{R}^{\mathbf{n}}\).

Problem 3: Compute all the homotopy groups of the \(n\)-torus \(T^{n}=S^{1} \times S^{1} \times \cdots \times S^{1}, n \geq 2\).

Problem 4: Consider the upper half space \(\mathbf{H}^{\mathbf{3}}=\{(x, y, z) \mid z>0\}\) equipped with hyperbolic metric \(g=\frac{d x^{2}+d y^{2}+d z^{2}}{z^{2}}\). Let \(P\) be the surface defined by \(\{z=x \tan \alpha, z>0\}\) for some \(\alpha \in\left(0, \frac{\pi}{2}\right)\). Compute the mean curvature of \(P\).

Problem 5: Suppose \(M\) is a compact 2-dimensional Riemannian manifold without boundary, with positive sectional curvature. Show that any two compact closed geodesics on \(M\) must intersect with each other.

Problem 6: Suppose \(\Sigma\) is a smooth compact embedded hypersurface (i.e. a codimension 1 submanifold) without boundary in \(\mathbf{R}^{\mathbf{n}}\) for \(n \geq 3\). Show that \(\Sigma\) is orientable.

2022 - Individual

Problem 1: The topological space \(X\) is obtained by gluing two tetrahedra as illustrated by the figure. There is a unique way to glue the faces of one tetrahedron to the other so that the arrows are matched. The resulting complex has 2 tetrahedra, 4 triangles, 2 edges and 1 vertex. Show that \(X\) can not have the homotopy type of a compact manifold without boundary.

Problem 2: Suppose \((M, h)\) is a closed (i.e., compact without boundary) Riemannian manifold, and \(h\) is a metric on \(M\) with \(\sec (h) \leq-1\), where \(\sec (h)\) is the sectional curvature. Suppose \(\Sigma\) is a closed minimal surface with genus \(g\) in \((M, h)\). Show that \(\operatorname{Area}(\Sigma) \leq 4 \pi(g-1)\). Remark: A minimal surface is an immersed surface with constant mean curvature 0.

Problem 3: For any topological space \(X\), the \(n\)-th symmetric product of \(X\) is the quotient of the Cartesian product \((X)^{n}\) by the action of the symmetric group \(S_{n}\), which permutes the factors in \((X)^{n}\). This space is denoted by \(\mathrm{SP}^{n}(X)\), and the topology is the natural quotient topology induced from \((X)^{n}\). Show that \(\mathrm{SP}^{n}\left(\mathbf{C P}^{1}\right)\) is homeomorphic to \(\mathbf{C P}^{n}\). Here \(\mathbf{C P}^{1}\) and \(\mathbf{C P}^{n}\) are equipped with the manifold topology.

Problem 4: Let \(M\) be a complete noncompact Riemannian manifold. \(M\) is said to have the geodesic loops to infinity property if for any \([\alpha] \in \pi_{1}(M)\) and any compact subset \(K \subset M\), there is a geodesic loop \(\beta \subset M \backslash K\), such that \(\beta\) is homotopic to \(\alpha\). Show that if a complete noncompact Riemannian manifold \(M\) does not have the geodesic loops to infinity property, then there is a line in the universal cover \(\tilde{M}\). Remark: A line is a geodesic \(\gamma:(-\infty, \infty) \rightarrow M\) such that \(\operatorname{dist}(\gamma(s), \gamma(t))=|s-t|\); a geodesic loop is a curve \(\beta:[0,1] \rightarrow M\) that is a geodesic and \(\beta(0)=\beta(1)\).

Problem 5: A topological space \(X\) is called an \(H\)-space if there exist \(e \in X\) and \(\mu: X \times X \rightarrow X\) such that \(\mu(e, e)=e\) and the maps \(x \rightarrow \mu(e, x)\) and \(x \rightarrow \mu(x, e)\) are both homotopic to the identity map. (a) Show that the fundamental group of an H-space is Abelian. (b) Show that the sphere \(S^{2022}\) is not an H-space. Historic Remark: “H” was suggested by Jean-Pierre Serre in recognition of the contributions in Topology by Heinz Hopf.

Problem 6: A hypersurface \(\Sigma \subset \mathbf{R}^{\mathbf{n + 1}}\) is called a shrinker if it satisfies the equation \(H(x)=\frac{1}{2}\langle x, \vec{n}\rangle\) Here \(H\) is the mean curvature, which is \(-\left\langle\operatorname{tr}_{A}, \vec{n}\right\rangle\) where \(A\) is the second fundamental form, \(x\) is the position vector, and \(\vec{n}\) is outer unit normal vector. (a) Show that \(S^{n}(\sqrt{2 n})\), the sphere with radius \(\sqrt{2 n}\), is a shrinker. (b) Show that any compact shrinker without boundary must intersect with \(S^{n}(\sqrt{2 n})\).

2023 - Individual

Problem 1: (a) Let \(G\) be a Lie group, \(\mathfrak{g}\) be its Lie algebra. The Maurer-Cartan form on \(G\) is the unique left-invariant \(\mathfrak{g}\)-valued 1-form such that \(\left.\omega\right|_{e}: T_{e} G \rightarrow \mathfrak{g}\) is the identity map, where \(e\) is the identity in \(G\). Show that the Maurer-Cartan form \(\omega\) satisfies the Maurer-Cartan equation \(d \omega+\frac{1}{2}[\omega, \omega]=0\). (b) Let \(G\) be a matrix group \(G L(n, \mathbb{R})\), give detailed computation to find the Maurer-Cartan form \(\omega\). (c) Let \(S E(3)\) be the special Euclidean group, i.e. it contains all transformations of \(\mathbb{R}^{3}\) (as 3-dim Euclidean space) of the form \(\mathbf{x} \mapsto t+A \mathbf{x}\), where \(\mathbf{t} \in \mathbb{R}^{\mathbf{3}}\) and \(A \in S O(3)\). Find an expression of the Maurer Cartan form \(\omega\) of \(S E(3)\), also check it satisfies the standard Cartan structure equations in Eulidean space.

Problem 2: Let \(M, N\) be closed, connected, oriented 3-manifolds with the first fundamental groups \(\pi_{1}\left(M_{1}\right)=\mathbb{Z}_{3} \oplus \mathbb{Z}^{2}, \quad \pi_{1}\left(M_{2}\right)=\mathbb{Z}_{6} \oplus \mathbb{Z}^{3}\). (a) Find all homology groups \(H_{n}\left(M_{1}, \mathbb{Z}\right)\) and \(H_{n}\left(M_{2}, \mathbb{Z}\right)\). (b) Find all homology groups \(H_{n}\left(M_{1} \times M_{2}, \mathbb{Q}\right)\). (c) Does there exist a closed connected oriented 3-manifold \(M\) with \(\pi_{1}(M)=\mathbb{Z}_{3} \oplus \mathbb{Z}^{2} \quad \text { or } \quad \pi_{1}(M)=\mathbb{Z}_{6} \oplus \mathbb{Z}^{3}\)?

Problem 3: (a) Let \(f\) be a diffeomorphism goup of a circle \(S^{1}\), assume \(f\) has no fixed point and it is generated by a smooth vector field, show that \(f\) must be conjugate to a rotation. (b) Show that there is a diffeomorphism \(f: S^{1} \rightarrow S^{1}\), such that \(f\) can not be generated by a smooth vector field but it is arbitrarily closed the identity map \(i: S^{1} \rightarrow S^{1}\) in \(C^{\infty}\)-topology.

Problem 4: (a) State the Leray-Hirsh theorem. (b) Let \(F l_{k}\left(\mathbb{C}^{n}\right)=\left\{\text { all } \mathrm{k} \text {-flags in } \mathbb{C}^{n}\right\}=\left\{\left(F_{0}, \cdots, F_{k}\right) \mid F_{i} \text { is an } i \text {-dim subspace of } \mathbb{C}^{n}, \text { s.t. } F_{0} \subset F_{1} \subset \cdots \subset F_{k} \subset \mathbb{C}^{n}\right\}\). Let \(\Phi: F l_{k}\left(\mathbb{C}^{n}\right) \rightarrow F l_{k-1}\left(\mathbb{C}^{n}\right)\) be the projection map sending a \(k\)-flag \(\left(F_{0}, \cdots, F_{k}\right)\) to a \((k-1)\)-flag \(\left(F_{0}, \cdots, F_{k-1}\right)\), it is known this is a fiber bundle. What is the fiber of \(\Phi\)? (c) Compute the Euler Characteristic \(\chi\left(F l_{n}\left(\mathbb{C}^{n}\right)\right)\).

Problem 5: On the Euclidean space \(\mathbb{R}^{n}\), we consider an \(n-1\) form \(\alpha\), which is of class \(C^{1}\), such that both \(\alpha\) and \(d \alpha\) are in \(L^{1}\). Show that \(\int_{\mathbb{R}^{n}} d \alpha=0\).

Problem 6: A complete Riemannian metric \(g_{i j}\) on a smooth manifold \(M^{n}\) is called a gradient expanding Ricci soliton if there exists a smooth function \(f\) on \(M^{n}\) such that the Ricci tensor Ric of the metric \(g\) is given by \(\text { Ric }+ \text { Hess } f=\lambda g\) for some negative constant \(\lambda<0\). Show that if \(M\) is compact, then a gradient expanding Ricci soliton must be an Einstein metric.

2024 - Individual

Problem 1: Let \(n>1\) be a positive integer. (i) Does there exist a map \(f: S^{2 n} \rightarrow \mathbb{C P}^{n}\) with \(\operatorname{deg}(f) \neq 0\)? Construct an example or disprove it. (ii) Does there exist a map \(f: \mathbb{C P}^{n} \rightarrow S^{2 n}\) with \(\operatorname{deg}(f) \neq 0\)? Construct an example or disprove it.

Problem 2: Let \(\Sigma \subset \mathbb{R}^{3}\) be an embedded surface in \(\mathbb{R}^{3}\). A surface is called minimal if, for any \(p \in \Sigma\), we have \(\kappa_{1}(p)+\kappa_{2}(p)=0\), where \(\kappa_{1}(p)\) and \(\kappa_{2}(p)\) are the two principal curvatures at \(p\). Prove that if \(\Sigma\) is closed, then \(\Sigma\) cannot be minimal.

Problem 3: Let \(M\) be a closed, simply connected 6-dimensional manifold. Suppose \(H_{2}(M)=\mathbb{Z}_{2}\). Prove that the Euler characteristic \(\chi(M) \neq-1\).

Problem 4: Let \((M, g)\) be a closed oriented \(n\)-dimensional Riemannian manifold. Let \(p \in M\) and \(\operatorname{Ric}_{p}\) be the Ricci curvature tensor at \(p,{ }_{p}\) be the scalar curvature at \(p\) which is defined to be \(S_{p}:=\frac{1}{n} \operatorname{Tr}_{g}\left(\operatorname{Ric}_{p}\right)\). Prove that the scalar curvature \(S(p)\) at \(p \in M\) is given by \(S_{p}=\frac{1}{\omega_{n-1}} \int_{S^{n-1}} \operatorname{Ric}_{p}(V, V) d S^{n-1}\) where \(\omega_{n-1}\) is the area of the unit sphere \(S^{n-1}\) in \(T_{p} M, V \in S^{n-1}\) are unit vector fields, and \(d S^{n-1}\) is the area element on \(S^{n-1}\).

Problem 5: Let \(S^{n}\) be the \(n\)-dimensional sphere with \(n \geq 2\), and let \(G\) be a finite group that acts freely on \(S^{n}\). Suppose \(G\) is non-trivial. Then, (i) Compute the homotopy groups of the quotient space \(\pi_{i}\left(S^{n} / G\right)\) for \(0 \leq i \leq n\). (ii) Suppose \(n\) is even. Prove that \(G\) is isomorphic to \(\mathbb{Z}_{2}\). (iii) Suppose \(n\) is odd. Show that \(G\) cannot be isomorphic to \(\mathbb{Z}_{p} \times \mathbb{Z}_{p}\) for \(p\) a prime number.

Problem 6: Let \(M\) be a closed oriented Riemannian manifold, where \(g_{t}\) is a family of smooth Riemannian metrics smoothly depending on \(t \in(-\epsilon, \epsilon)\). Suppose there exists a family of eigenfunctions \(f_{t}\) and eigenvalues \(\lambda_{t}\) smoothly depending on \(t\) such that \(\Delta_{g_{t}} f_{t}=\lambda_{t} f_{t}\) where \(\Delta_{g_{t}}\) is the Laplace-Beltrami operator defined using the Riemannian metric \(g_{t}\). Additionally, assume that \(f_{0}\) is not a constant function. We define \(\dot{\lambda}:=\left.\frac{d}{d t}\right|_{t=0} \lambda_{t}\) and \(\dot{\Delta}:=\left.\frac{d}{d t}\right|_{t=0} \Delta_{g_{t}}\). Prove the following: (i) As \(\lambda_{0}\) is an eigenvalue of \(\Delta_{g_{0}}\), let \(V_{\lambda_{0}}:=\operatorname{Ker}\left(\Delta_{g_{0}}-\lambda_{0}\right)\) be the eigenspace of \(\lambda_{0}\), and let \(\Pi: L^{2}\left(M, g_{0}\right) \rightarrow V_{\lambda_{0}}\) be the orthogonal projection onto the eigenspace. Prove that \(\dot{\lambda}\) is an eigenvalue of the operator \(\Pi \circ \Delta^{\prime}: V_{\lambda_{0}} \rightarrow V_{\lambda_{0}}\). (ii) Let \(\varphi_{t}: M \rightarrow M\) be a 1-parameter family of diffeomorphisms of \(M\) and assume \(g_{t}=\varphi_{t}^{*} g_{0}\). Prove that \(\dot{\lambda}=0\).

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Amazing! You finished such a long list!