For shortest we will refer a smooth manifold simply as a manifold.

Prove that:

(a) The sphere is a manifold of dimension 2.

(b) The torus is a manifold of dimension 2.

(c) If $M$ and $N$ are manifolds, then $M\times N$ is a manifold.

(d) The homogenous space of dimension 2 is a manifold.

Show that:

(a) Every manifold is locally compact.

(b) Every manifold is locally path-connected.

(c) Every connected manifold is path-connected.

Prove that if $M$ is a path-connected manifold then given $p_1,p_2\in M$ there exists a differentiable map from $p_1$ to $p_2$.

Remembering the definition of differentiable map:

Let $S(n):={M\in M_n(\mathbb{R})| M^t=M}$. Prove that $S(n)$ is a manifold of dimension $n(n+1)/2$. Show that the function is differentiable and compute $df_M$ for any $M\in M_n(\mathbb{R})$. In particular compute $df_{\operatorname{Id}_n}$.

Let $(\phi, X)$ and $(\psi, Y)$ two local charts of $p\in M$. Then we know that $\lbrace \frac{\partial}{\partial x^i}|_p\rbrace$ and $\lbrace \frac{\partial}{\partial y^i}|_p\rbrace$i are basis of $T_pM$. Compute the coeficients of the basis change matrix.

Show that a subset $X$ of a manifold $M$ is a submanifold of the same dimension if and only if is open.

Show that given a manifold $M$ of dimension $n$, el haz tangente $TM$ es una manifold of dimension $2n$.

Given two smooth manifolds $M,N$ compute $T_{(p,q)}(M\times N)$. In addition compute the tangent space of a point of a $\mathbb{R}$-vector space of finite dimension.