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I'm reading do Carmo's book, Riemannian geometry and I don't understand some things at the page 126.

  1. I know that on a Riemannian manifold, $(\nabla_X Y)_p$ depends on $X^i_p,Y^i_p$ and $(\partial_iY^i)_p$ if locally $X=X^i\partial_i$ and $Y=Y^i\partial_i.$

  2. In the case when we have $f:M\to\overline{M}$ immersion we have $\nabla_X Y =(\overline{\nabla}_\overline{X}\overline{Y})^T$ with $\overline{X}$ and $\overline{Y}$ are extension of $X$ and $Y.$ my first question is $\overline{X}$ is a field on $\overline{M}$ such that $\overline{X}=X$ on $ M$?

  3. We define $B(X,Y)=\overline{\nabla}_\overline{X}\overline{Y}-\nabla_X Y.$ this is easy to see that $B$ is correct, bi linear and symmetric, but do Carmo says that $B(X,Y)_p$ depend only on $X_p$ and $Y_p.$ My question is way?

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  • $\begingroup$ About 2, yes $\bar{X}$ is a vector field on $\bar{M}$; In terms of 3, first notice that $B(X,Y)$ is independent of extension of $X$ since $\bar{\nabla}_{\bar{X}}\bar{Y}$ is, then by symmetry, it is also independent of extension of $Y$. $\endgroup$ – H-H Aug 8 '18 at 17:37

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