Prove that the sets $$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2+x_3^2=x_4^2 \} , $$ $$ \{ \left( x_1,x_2,x_3,x_4 \right)\in\mathbb{R}^4|x_1^2+x_2^2=x_3^2+x_4^2 \}, $$ are not homeomorphic on the Euclidean topology induced on them.

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

Let ( M , g ) be a Riemannian manifold. Extend the metric tensor g to the set of tangents TM with the following specification: if $a,b\in T_v TM \, (v\in T_p M)$, then $$\tilde g_v (a, b): = g_p (\dot {\alpha} (0), \dot {\beta} (0) ) + g_p (D _ {\alpha} X(0) , D_{\beta} Y(0) )$$ where $\alpha, \beta$ are curves in M such that $\alpha(0) = \beta(0) = p$. X and Y are vector fields along $\alpha,\beta$ respectively, with the condition $\dot X (0) = a,\dot Y(0) = b$. $D _{\alpha}$ and $D _{\beta}$ are the operators of the covariant derivative along the corresponding curves according to the Levi-Civita connection. Is the eigenfunction from the Riemannian manifold (M,g) to the Riemannian manifold $(TM, \tilde g)$ harmonic?

Let $M = F_1\times\cdots\times F_k$ be the product of $k$ smooth, closed surfaces (2-dimensional, $C^\infty$, compact, connected, manifold without boundary), $s$ of which are non-orientable. Prove that $M$ can be embedded in $\mathbb{R}^{2k+s+1}$.

Let $F^2$ be a closed, oriented 2-dimensional smooth surface, $f : F^2 \to F^2$ is a smooth homeomorphism whose order is an odd prime p (i.e., the p-th iterate $f \circ f \circ \cdots \circ f$ is the identity). Then f has a finite number of fixed points: $P_1 , ..., P_s$. In the tangent plane at the fixed point $P_i$, a positively directed (i.e., compatible with the direction of the surface) base can be chosen in which f is differentiated by a rotation with positive angle $2\pi k_i/p$ , where $k_i$ is a natural number, $0 < k_i < p$ . Prove that $$\sum_{i = 1}^s k_i^{p-2}\equiv0\pmod{p}$$

Let $ p,q $ be the two most distant points (in the Euclidean sense) of a closed surface $ M $ embedded in the Euclidean space. [b]a)[/b] Show that the tangent planes of $ M $ at $ p $ and $ q $ are parallel. [b]b)[/b] What happened if $ M $ would be a closed curve of $ \mathcal{C}^{\infty } \left(\mathbb{R}^3\right) $ class, instead?

Let $ U\subset\mathbb R^n$ be an open set, and let $ L: U\times\mathbb R^n\to\mathbb R$ be a continuous, in its second variable first order positive homogeneous, positive over $ U\times (\mathbb R^n\setminus\{0\})$ and of $ C^2$-class Langrange function, such that for all $ p\in U$ the Gauss-curvature of the hyper surface \[ \{ v\in\mathbb R^n \mid L(p,v) \equal{} 1 \}\] is nowhere zero. Determine the extremals of $ L$ if it satisfies the following system \[ \sum_{k \equal{} 1}^n y^k\partial_k\partial_{n \plus{} i}L \equal{} \sum_{k \equal{} 1}^n y^k\partial_i\partial_{n \plus{} k} L \qquad (i\in\{1,\dots,n\})\] of partial differetial equations, where $ y^k(u,v) : \equal{} v^k$ for $ (u,v)\in U\times\mathbb R^k$, $ v \equal{} (v^1,\dots,v^k)$.

Prove that for a $2$-dimensional Riemannian manifold there is a metric linear connection with zero curvature if and only if the Gaussian curvature of the Riemannian manifold can be written as the divergence of a vector field.