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Let $f: M \to N$ be an immersion of a differentiable manifold $M$ into a Riemannian manifold $N$. Assume that $M$ has the Riemannian metric induced by $f$. Let $p \in M$ and let $U \subset М$ be a neighborhood of $p$ such that $f(U) \subset N$ is a submanifold of $M$. Further, suppose that $X, Y$ are differentiable vector fields on $f(U)$ which extend to differentiable vector fields $X^*, Y^*$ on an open set of $N$. Define $$(\nabla_x Y)(p) =\text{ tangential component of }(\overline{\nabla}_{x^*} Y^*)(p),$$ where $\overline{\nabla}$ is the Riemannian connection of $N$. Prove that $\nabla$ is the Riemannian connection of $M$.

For the bounty: I need a detailed explanation of the proof that the connection is compatible with the metric. [this will be removed as soon as the bounty expires]

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    In the future, please cite where you got this problem, and what your motivation is for asking. For instance, this is Exercise 3 in Chapter 2 of do Carmo's "Riemannian Geometry." My own professor actually just assigned this problem to us as homework last week. (Also, welcome to math.stackexchange!)2011-10-01
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    At any rate, you know the Riemannaian connection on $M$ is unique, so you only need to check that $\nabla$ is metric compatible and symmetric.2011-10-01
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    Excuse me, is my first time2011-10-01
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    One more thing: if possible, try to use Latex syntax when writing in order to make your text more legible. Your next texts will appear in a very nice fashion.2012-04-21
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    @JesseMadnick I would appreciate if you take a look at my comments. I do not know why I cannot call you from your answer, so I call you from here. Thank you.2015-03-24
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    "$f(U)\subset N$ is a submanifold of N" means an embedding?2018-12-06

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