\begin{tikzpicture}[scale=\textwidth/10cm,samples=200]
    \draw[thick] (0,0) arc (0:360: 4cm and 2cm);
    \draw[solid] (-2,.5) arc (0:-180: 2cm and .8cm);
    \draw[solid] (-4,.5) arc (90:150: 2cm and .8cm);
    \draw[solid] (-4,.5) arc (90:30: 2cm and .8cm);
    \draw  (-4,-3) node {$T^2 = S^1 \times S^1$};
    \draw[red] (-4,.5) ellipse (3cm and 1cm); 
	\draw[red] (-5,-.25) arc (90:270:.5cm and .85cm);
	\draw[red, dashed] (-5,-.25) arc (90:-90:.5cm and .85cm);
\end{tikzpicture}