(a) Let $X \subset \mathbb{R}^{N}$ be a manifold. Give the definition of the tangent space $T_{p} X$ of $X$ at a point $p \in X$.

(b) Show that $X:=\left\{-x_{0}^{2}+x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=-1\right\} \cap\left\{x_{0}>0\right\}$ defines a submanifold of $\mathbb{R}^{4}$ and identify explicitly its tangent space $T_{\mathbf{x}} X$ for any $\mathbf{x} \in X$.

(c) Consider the matrix group $O(1,3) \subset \mathbb{R}^{4^{2}}$ consisting of all $4 \times 4$ matrices $A$ satisfying

$$A^{t} M A=M$$

where $M$ is the diagonal $4 \times 4$ matrix $M=\operatorname{diag}(-1,1,1,1)$.

(i) Show that $O(1,3)$ forms a group under matrix multiplication, i.e. it is closed under multiplication and every element in $O(1,3)$ has an inverse in $O(1,3)$.

(ii) Show that $O(1,3)$ defines a 6-dimensional manifold. Identify the tangent space $T_{A} O(1,3)$ for any $A \in O(1,3)$ as a set $\{A Y\}_{Y \in \mathfrak{S}}$ where $Y$ ranges over a linear subspace $\mathfrak{S} \subset \mathbb{R}^{4^{2}}$ which you should identify explicitly.

(iii) Let $X$ be as defined in (b) above. Show that $O^{+}(1,3) \subset O(1,3)$ defined as the set of all $A \in O(1,3)$ such that $A \mathbf{x} \in X$ for all $\mathbf{x} \in X$ is both a subgroup and a submanifold of full dimension.

[You may use without proof standard theorems from the course concerning regular values and transversality.]