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\begin{document}
\title{{\bf  Homework set 4 - due 10/27/20}\\ {\large Math 5047 \---  Renato Feres} 
}
\author{}
\date{}  
\maketitle
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 Turn in problems 1, 2, 3, 4.
\begin{enumerate}
\item {\bf Example of a geodesically incomplete connection.}  We consider an affine connection $\nabla$ on the (tangent bundle of the) $1$-dimensional torus, $\mathbb{T}$. Equivalently,
we may work  in $\mathbb{R}$ and assume that the single Christoffel symbol $\Gamma(x)$,
defined by $$\nabla_{\frac{d}{dx}}\frac{d}{dx}=\Gamma(x)\frac{d}{dx}, $$
 is periodic: $\Gamma(x+1)=\Gamma(x)$. Let $x(t)$ be a geodesic relative to $\nabla$ with initial conditions $x(0)=0$ $\dot{x}(0)=1$, where $\dot{x}(t)$ is the derivative in $t$. 
 \begin{enumerate}
\item Show that $x(t)$ satisfies the differential equation
 $$\ddot{x} +\Gamma(x)\dot{x}^2=0. $$
\item Suppose that $\Gamma(x)=a$ is a constant. Find an equation for $x(t)$. 
\item Assuming $\Gamma(x)=a$, for which values of $a$ is $\nabla$ geodesically complete? (That is, for which $a$ are geodesics  defined for all time $t$?)
\item Now suppose the torus $\mathbb{T}$ is equipped with a Riemannian  metric and that $\nabla$ (for a not necessarily constant $\Gamma(x)$) is
a metric connection.   Let us write $$\left\langle \frac{d}{dx}, \frac{d}{dx} \right\rangle=g(x)$$
for the single metric coefficient, where $g(x)$ is periodic: $g(x+1)=g(x)$. The metric connection condition means
$$\frac{d}{dx}g(x) = \frac{d}{dx}\left\langle \frac{d}{dx}, \frac{d}{dx} \right\rangle = 2\left\langle \nabla_{\frac{d}{dx}}\frac{d}{dx}, \frac{d}{dx} \right\rangle =2\Gamma(x) g(x).  $$
Show that $$ \int_0^1 \Gamma(x)\, dx =0.$$ Also show that $x(t)$ satisfies the differential equation
\begin{equation}\label{geodesic}\dot{x}(t)\sqrt{g(x(t))}=\text{const.}= \sqrt{g(0)}. \end{equation}
In this case, as $g(x)$ is periodic, hence bounded and bounded away from $0$, it is not difficult to show that $x(t)$ is defined for all $t$. Hence, in the metric case, $\nabla$ is geodesically complete.   Note: for any value of $x\in \mathbb{R}$, let $t(x)$ be given by the following equation, obtained by integrating
Equation (\ref{geodesic}):
$$t(x)=\int_0^x \sqrt{\frac{g(s)}{g(0)}}ds. $$
It is clear from this equation that $t$ can take any real value.
\end{enumerate}




\item {\bf Riemannian metric on projective space $\mathbb{R}P^n$.}
\begin{enumerate}
\item Prove that the antipodal mapping $A:S^n\rightarrow S^n$ given by
$A(p)=-p$ is an isometry  of $S^n=\left\{p\in \mathbb{R}^{n+1}:|p|=1\right\}$. 
\item Use this fact to introduce a Riemannian metric  on the real projective space $P^n(\mathbb{R})$ such that
the natural projection $\pi:S^n\rightarrow P^n(\mathbb{R})$ is a local isometry.
\end{enumerate}





\item {\bf Isometric immersion of the flat torus in Euclidean space.} Obtain an isometric immersion  of the flat torus $\mathbb{T}^n$ into $\mathbb{R}^{2n}$.
Suggestion: start with $n=1$ and recall that $\mathbb{T}^n$ is the Cartesian product of $n$ copies of $\mathbb{T}^1$. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item {\bf Conformally Euclidean metrics.} Let $\eta(\mathbf{x})$ be a positive function defined on an open subset $\mathcal{U}$ in $\mathbb{R}^n$ and define the Riemannian metric on $\mathcal{U}$
$$ \langle u,v\rangle_x := \eta^2(x)u\cdot v.$$
We say that the metric is conformally Euclidean.  
\begin{enumerate}
\item Express the Riemannian volume of a compact subset $A\subseteq\mathcal{U}$ as an integral over $A$. 
\item Express the length of a differentiable curve $c(t)\in \mathcal{U}$, $t\in [a,b]$, as an ordinary integral over the interval $[a,b]$. 
\end{enumerate}

 
 

\item   {\bf Geodesics of a surface of revolution.}
Denote by $(u,v)$ the Cartesian  coordinates of $\mathbb{R}^2$.  Define the function 
$\varphi:U\subseteq \mathbb{R}^2\rightarrow \mathbb{R}^3$ given by $$\varphi(u,v)=(f(v)\cos u, f(v)\sin u, g(v))$$
where $U=(u_0,u_1)\times (v_0,v_1)$, $f$ and $g$ are differentiable functions with ${f'(v)}^2+{g'(v)}^2 \neq 0$ and
$f(v)\neq 0$. 
\begin{enumerate}
\item Show that $\varphi$ is an immersion.
(The image $\varphi(U)$ is the surface generated by the rotation of the curve  $(f(v),g(v))$ around
the axis $0z$ and is called a {\em surface of revolution} $S$. The image by $\varphi$ of the curves $u=$ constant and $v=$ constant
are called {\em meridians} and {\em parallels}, respectively, of $S$.) 
\item Show that the induced metric in the coordinates $(u,v)$ is given by
$$g_{11}=f^2, \ \ \ g_{12}=0, \ \ \ g_{22}=f'^2 +g'^2. $$
\item Show that local equations of a geodesic $\gamma$ are
\begin{align*}
\frac{d^2u}{dt^2}+\frac{2ff'}{f^2}\frac{du}{dt}\frac{dv}{dt}&=0\\
\frac{d^2v}{dt^2}-\frac{ff'}{f'^2 +g'^2}\left(\frac{du}{dt}\right)^2 +\frac{f'f''+g'g''}{f'^2 +g'^2}\left(\frac{dv}{dt}\right)^2&=0.
\end{align*}
\item Obtain the following geometric meaning of the equations above: the second equation is, except for meridians and parallels, equivalent to the fact that 
the ``energy'' $|\gamma'(t)|^2$ of a geodesic is constant along $\gamma$; the first equation signifies that if $\beta(t)$ is the oriented angle, $\beta(t)<\pi$, of $\gamma$
with a parallel $P$ intersecting $\gamma$ at $\gamma(t)$, then $$r\cos\beta = \text{ constant}, $$
where $r$ is the radius of the parallel  $P$. (The equation above is called {\em Clairaut's relation}.)
\item (You don't have to do this part. If you are curious, see do Carmo's  {\em Differential Geometry of Curves and Surfaces}, Dover Publications, 2016, page 262.) Use Clairaut's relation to show that a geodesic of the paraboloid 
$$ \left(f(v)=v, g(v)=v^2, v>0, -\epsilon<u<2\pi+\epsilon\right)$$
which is not  a meridian, intersects itself an infinite number of times. (See Figure 6 on page 79.)
\end{enumerate}




\end{enumerate} 
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