\begin{tikzpicture}
    \draw (0,0) ellipse (2cm and 1cm);
    \begin{scope}[xshift=-2.2cm,yshift=-2.8cm]
        \draw[->] (0,0) -- (0,1.5);
        \draw[->] (0,0) -- (1.5,0);
        \node at (0.3,0.16) {$\mathbb{R}^n$};
    \end{scope}
    \begin{scope}[xshift=0.4cm,yshift=-2.8cm]
        \draw[->] (0,0) -- (0,1.5);
        \draw[->] (0,0) -- (1.5,0);
        \node at (0.3,0.16) {$\mathbb{R}^n$};
    \end{scope}
    \def\ringa{(-0.3,0) circle (0.5cm)}
    \def\ringb{(+0.3,0) circle (0.5cm)}

    \begin{scope}[even odd rule]
        \clip \ringa;
        \fill[pattern color=red,pattern=north east lines] \ringb;
    \end{scope}

    \begin{scope}[even odd rule,shift={(-0.7,-2)}]
        \clip \ringa;
        \fill[draw=red,pattern color=red,pattern=north east lines] \ringb;
    \end{scope}

    \begin{scope}[even odd rule,shift={(+0.7,-2)}]
        \clip \ringb;
        \fill[draw=red,pattern color=red,pattern=north east lines] \ringa;
    \end{scope}
    \draw \ringa;
    \draw \ringb;

    \node at (-1,0.3) {$U_i$};
    \node at (+1,0.3) {$U_j$};
    \node at (-1.9,-2) {$V_i$};
    \node at (+1.9,-2) {$V_j$};
    \node at (+2.0,0.7) {$X$};

    \path[->] (-0.35,0)  edge [bend angle=10,bend right] node[label={[label distance=0.1cm]210:$\varphi_i$}] {} (-1,-1.5);
    \path[->] (+0.35,0)  edge [bend angle=10,bend left]  node[label={[label distance=0.1cm]-30:$\varphi_j$}] {} (+1,-1.5);

    \draw (-1,-2) circle (0.5cm);
    \draw (+1,-2) circle (0.5cm);

    \draw[->, red, thick] (-0.3,-2) -- (0.3,-2);
\end{tikzpicture}