1.Let $X$ be the quotient space of $\bbS^2$ under the identifications $x\sim -x$ for $x$ in the equator $\bbS^1$. Compute the homology groups $H_n(X)$. Do the same for $\bbS^3$ with antipodal points of the equator $\bbS^2\subset\bbS^3$ identified.
2.Let $M\to\bbR^3$ be a graph defined by $z=f(u,v)$ where $\sed{u,v,z}$ is a Descartes coordinate system in $\bbR^3$. Suppose that $M$ is a minimal surface. Prove that
(1) The Gauss curvature $K$ of $M$ can be expressed as $$\bex K=\lap \ln \sex{1+\frac{1}{W}},\quad W=\sqrt{1+\sex{\frac{\p f}{\p u}}^2+\sex{\frac{\p f}{\p v}}^2}. \eex$$
(2) If $f$ is defined on the whole $uv$-plane, then $f$ is a linear function (Bernstein theorem).
3. Let $M=\bbR^2/ \bbZ^2$ be the two-dimensional torus, $L$ the line $3x=7y$ in $\bbR^2$, and $S=\pi(L)\subset M$ where $\pi:\bbR^2\to M$ is the projection map. Find a differential form on $M$ which represents the Poincar\'ee dual of $S$.
4. Let $(\tilde M,\tilde g)\to (M,g)$ be a Riemannian submersion. This is a submersion $p:\tilde M\to M$ such that for each $x\in \tilde M$, $\ker^\perp (Dp)\to T_{p(x)}M$ is a linear isometry.
(1) Show that $p$ shortens distance.
(2) If $(\tilde M,\tilde g)$ is complete, so is $(M,g)$.
(3) Show by example that if $(M,g)$ is complete, $(\tilde M,\tilde g)$ may not be complete.
5. Let $\psi:M\to\bbR^3$ be an isometric immersion of a compact surface $M$ into $\bbR^3$. Prove that $$\bex \int_M H^2\rd \sigma \geq 4\pi, \eex$$ where $H$ is the mean curvature of $M$ and $\rd \sigma$ is the area element of $M$.
6. The unit tangent bundle of $\bbS^2$ is the subset $$\bex T^1(\bbS^2)=\sed{(p,v)\in\bbR^2;\ \sen{p}=1,\ (p,v)=0, \sen{v}=1}. \eex$$ Show that it is a smooth submanifold of the tangent bundle $T(\bbS^2)$ of $\bbS^2$ and $T^1(\bbS^2)$ is diffeomorphic to $\bbR P^3$.