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.

Let $\Sigma \subset \mathbb{R}^3$ be an embedded surface in $\mathbb{R}^3$. A surface is calle $\mathrm{d}$ minimal if, for any $p \in \Sigma$, we have $\kappa_1(p)+\kappa_2(p)=0$, whe re $\kappa_1(p)$ and $\kappa_2(p)$ are the two principal curvatures at $p$. Prov e that if $\Sigma$ is closed, then $\Sigma$ cannot be minimal.

Let $M$ be a closed, simply connected 6-dimensional manifol d. Suppose $H_2(M)=\mathbb{Z}_2$. Prove that the Euler characteristic $\chi(M) \neq-1$.

Let $(M, g)$ be a closed oriented $n$-dimensional Riemannian manifold. Let $p \in M$ and $\operatorname{Ric}_p$ be the Ricci curvature tensor a $\mathrm{t} p, p$ be the scalar curvature at $p$ which is given by 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 eleme nt on $S^{n-1}$.

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.

Let $M$ be a closed oriented Riemannian manifold, where $g_t$ is a family of smooth Riemannian metrics smoothly depending on $t \in(-\varepsilon, \varepsilon)$. Suppose there exists a family of eigenfunctio ns $f_t$ 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 co nstant 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 th e 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 diffeomorph isms of $M$ and assume $g_t=\varphi_t^* g_0$. Prove that $\dot{\lambda}=0$.