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OK, this is more of a spot for an elaboration on a question I just posted in another thread. Not quite duplicating threads, I hope, I just wanted to have this not buried in another spot...

So, the question is this:

Let's say that we have a smooth manifold [tex]\mathcal{M}[/tex]that may be viewed as a surface in [tex]\mathbb{R}^3[/tex] given by an embedding [tex]x[/tex]. Denote by [tex]n[/tex] the outward-pointing unit normal to [tex]\mathcal{M}[/tex] (yes, we are assuming that [tex]\mathcal{M}[/tex] is orientable), and by [tex]g[/tex] and [tex]h[/tex] the metric and second fundamental form, resp., defined by

[tex]g_{ij}=\left<\dfrac{\partial x}{\partial u^i},\dfrac{\partial x}{\partial u^j}\right>[/tex]

[tex]h_{ij}=-\left<\dfrac{\partial^2x}{\partial u^i\partial u^j},n\right>[/tex]

with respect to some local coordinates [tex]\left\lbrace u^1,u^2\right\rbrace[/tex] for some (open) region of [tex]\mathcal{M}[/tex].

What I need to understand, basically, is how this notation works. I can easily see how one would obtain

[tex]g_{ij}=\dfrac{\partial y^k}{\partial u^i}\dfrac{\partial y^k}{\partial u^j};\hspace{0.75cm}k\text{ summed}[/tex]

from the definiton that [tex]g=g_{ij}du^i\otimes du^j[/tex], [tex]y^k=y^k\left(u^1,u^2\right)[/tex], but I don't see how to go from here to the pairing above, and am even more confused about the pairing used to define [tex]h_{ij}[/tex] above.

Any thoughts?

So, the question is this:

Let's say that we have a smooth manifold [tex]\mathcal{M}[/tex]that may be viewed as a surface in [tex]\mathbb{R}^3[/tex] given by an embedding [tex]x[/tex]. Denote by [tex]n[/tex] the outward-pointing unit normal to [tex]\mathcal{M}[/tex] (yes, we are assuming that [tex]\mathcal{M}[/tex] is orientable), and by [tex]g[/tex] and [tex]h[/tex] the metric and second fundamental form, resp., defined by

[tex]g_{ij}=\left<\dfrac{\partial x}{\partial u^i},\dfrac{\partial x}{\partial u^j}\right>[/tex]

[tex]h_{ij}=-\left<\dfrac{\partial^2x}{\partial u^i\partial u^j},n\right>[/tex]

with respect to some local coordinates [tex]\left\lbrace u^1,u^2\right\rbrace[/tex] for some (open) region of [tex]\mathcal{M}[/tex].

What I need to understand, basically, is how this notation works. I can easily see how one would obtain

[tex]g_{ij}=\dfrac{\partial y^k}{\partial u^i}\dfrac{\partial y^k}{\partial u^j};\hspace{0.75cm}k\text{ summed}[/tex]

from the definiton that [tex]g=g_{ij}du^i\otimes du^j[/tex], [tex]y^k=y^k\left(u^1,u^2\right)[/tex], but I don't see how to go from here to the pairing above, and am even more confused about the pairing used to define [tex]h_{ij}[/tex] above.

Any thoughts?

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