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\begin{center}
\begin{Large}
\textbf{Geometr\'{\i}a IV}
\\
\
\\
Examen final \qquad {\begin{normalsize}
7 de junio de 2007
                      \end{normalsize}}
\end{Large}
\end{center}

\ejer
Responder razonadamente a las siguientes preguntas:

\vskip2pt
a) Sean $\omega=(f(x)+y)dx+(g(y)+x)dy$ y $F:\R^2\longrightarrow
\R^2$, ?`es $F^*\omega$ necesariamente exacta?

\vskip2pt
b) ?`Cu\'{a}l es el n\'{u}mero m\'{a}ximo de componentes nulas que puede tener
una m\'{e}trica en una variedad de dimensi\'{o}n $n$?

\ejer
Hallar todas las componentes del tensor $T=\dfrac{\partial}{\partial
x}\otimes dy$ en $\R^2$ cuando se emplean coordenadas polares.

\ejer
Sea la subvariedad de $\R^3$ dada por $S=\{(x,y,z)\in\R^3\;:\;
(x-1)^2+y^2+z^2=R^2\}$ con $R>3$ y sea $\omega=(xdy\wedge
dz+ydz\wedge dx+zdx\wedge
dy)/\|\vec{x}\|^3\in\Omega^1(\R^3-\{\vec{0}\})$,
$\vec{x}=(x,y,z)$. Demostrar que si
$j:S\longrightarrow\R^3-\{\vec{0}\}$ es la inclusi\'{o}n, $\int_S
j^*\omega$ es constante, no depende de~$R$. \textit{Indicaci\'{o}n}:
Se puede dar por supuesto $d\omega=0$.


\ejer
Calcular las ecuaciones que definen las geod\'{e}sicas en el semiplano
$\R\times\R^+$ con la m\'{e}trica $y^{-2}dx^2+y^{-2}dy^2$.

\ejer
Supongamos en $\R^2$ una m\'{e}trica tal que se cumple
$\nabla_1(\partial_1+2\partial_2)
=\nabla_1(\partial_1+\partial_2)=\partial_1$
y
$\nabla_2(\partial_2)=0$.
Hallar los s\'{\i}mbolos de Christoffel y estudiar si la curva
$(x(t),y(t))=(\log(t+1),0)$ es una geod\'{e}sica.


\ejer
Responder razonadamente a las siguientes preguntas:

\vskip2pt
a) Si $V$ y $W$ son dos campos de vectores en $\R^2$, ?`es
$V_{;1}^1W_{;1}^1-V_{;2}^2W_{;2}^2$ independiente de la carta
elegida?

\vskip2pt
b) Si $f$ es una funci\'{o}n inyectiva entre variedades, ?`es siempre
$f^*$ una aplicaci\'{o}n inyectiva entre los grupos de cohomolog\'{\i}a
$H^1$?



\vfill
\begin{footnotesize}
\textbf{Nota:} La distribuci\'{o}n de la puntuaci\'{o}n es
$(1+1)+1.5+1.5+1.5+1.5+(1+1)$.
\end{footnotesize}

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