Riemannian geometry

The metric $g = dt^2 + t^4\, d\theta^2$ (for $t > 0$) has the form $dt^2 + f(t)^2 d\theta^2$. What is its Gaussian curvature at $t = 1$?

Let $M$ be a closed orientable surface of genus $3$ with any Riemannian metric. Compute $\dfrac{1}{\pi}\displaystyle\int_M K\, dA$.

Which of these curves on the unit sphere $S^2$, parametrised with constant speed, are geodesics? (Choose all that apply.)

The paraboloid $z = x^2 + y^2$, with the metric induced from $\R^3$, is complete and has Gaussian curvature $K > 0$ at every point, yet it is not compact. Which hypothesis of the Bonnet–Myers theorem fails?

In the proof of the fundamental theorem, the Koszul formula expresses $2\langle\nabla_X Y, Z\rangle$ in terms of the metric and brackets. Which property is needed to conclude that $\nabla_X Y$ itself is determined?

On the unit sphere, let $\gamma$ be the unit-speed great-circle arc of length $3\pi/2$ from $p$ to $q$. Which statement is true?

A geodesic triangle in the hyperbolic plane (curvature $-1$) has angles $\pi/3$, $\pi/4$ and $\pi/6$. What is its area?

In semicircle_geodesic, the curve is $x = \sinh t/\cosh t$, $y = 1/\cosh t$, with $x' = 1/\cosh^2 t$ and $y' = -\sinh t/\cosh^2 t$. Which conjunct states that it has unit speed for the hyperbolic metric $(dx^2 + dy^2)/y^2$?

Write excess(a, b, c) for three unit vectors $a, b, c \in \R^3$ (tuples) spanning a geodesic triangle on the unit sphere. Return $\alpha + \beta + \gamma - \pi$, where the angle at $a$ is the angle between the tangent vectors $b - \langle a, b\rangle a$ and $c - \langle a, c\rangle a$ (the initial directions of the great-circle arcs from $a$), and similarly at $b$ and $c$. By local Gauss–Bonnet the excess is the area.