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I would like to find some non-trivial and "geometric" examples of hermitian connections (that is compatible with a give hermitian metric) on complex hermitian vector bundles over a smooth real manifold. More specifically:

Let $M\subset\mathbb C^{n+1}$ be a smooth real hypersurface and consider its real tangent bundle $T_M^{\mathbb R}$ and its complex tangent bundle $T_M=T_M^{\mathbb R}\cap i T_M^{\mathbb R}$, where $i$ is the imaginary unit in $\mathbb C^{n+1}$. Let me call $g$ the standard flat riemmanian metric on $\mathbb C^{n+1}\simeq\mathbb R^{2n+2}$ and $h$ the standard flat hermitian metric on $\mathbb C^{n+1}$ (so that $g$ is just the real part of $h$). Finally, let $\pi\colon T_M^{\mathbb R}\to T_M$ be the orthogonal projection with respect to $g$.

Consider the Levi-Civita connection $\nabla_M$ on $T_M^{\mathbb R}$ with respect to (the restriction on $T_M^{\mathbb R}$ of) $g$ and call $h_M$ the restriction of $h$ to $T_M$. Then $(T_M,h_M)$ is a complex hermitian vector bundle on the smooth real manifold $M$. Next, set $D:=\pi\circ\nabla$.


Is $D$ a (complex) linear connection on $T_M$?

If so, is it a hermitian connection with respect to $h_M$?

Is it true at least when $M=S^{2n+1}$ is the unit sphere in $\mathbb C^{n+1}$?

Thanks in advance!

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