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In the Do Carmo book's Riemmanian Geometry say:

Observe that if $\phi:M\rightarrow M$ is a differential map, $v\in T_p M$ and $f$ is a real differentiable function in a neighborhood of $\phi(p)$, we have $$(d\phi(v)f)\phi(p)=v(f\circ \phi)(p) $$ Indeed, let $\alpha:(-\epsilon, \epsilon)\rightarrow M$ be a differentiable curve with $\alpha(0)=p$ and $\alpha'(0)=v$. The $$(d\phi(v)f)\phi(p)=\displaystyle\frac{d(f\circ\phi\circ\alpha)}{dt} \displaystyle =v(f\circ \phi)(p)$$ at $t=0$.

I do not understand how to interpret the first equation, I mean $d\phi(v)$ is a function such that sends real and derivable function on $M$ to a real number then $d\phi(v)f$ is a real number. And then $(d\phi(v)f)\phi(p)$ what is?

I know that I am misunderstanding something. Somebody can clarify me this equation?

Thanks!

EQJ
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  • $d\phi(v)$ is a tangent vector in $T_{\phi(p)} M$. Tangent vectors act on functions. I would probably use the notation $(((d\phi)(\phi(p)))(v))(f)$ which is maybe less and maybe more confusing. Or just $((d\phi)(v)), f$. The result is a number. – Peter Franek Mar 11 '15 at 12:59
  • Can you explain me step by step (parenthesis by parenthesis) how this function are applied? – EQJ Mar 11 '15 at 13:07
  • I understand the following $d\phi(\phi(p))$ is the differential at the point $\phi(p)$ or $d_{\phi(p)}(\phi)$. Then $d_{\phi(p)}(\phi)(v)$ is a map that send function to real. And $d_{\phi(p)}(\phi)(v)(f)$ is a real. Am I right? Is this the same meaning of the notation used by Do Carmo? – EQJ Mar 11 '15 at 13:21
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    Yes, right. $d \phi$ takes vectors to vectors and a vectors maps smooth function to reals. – Peter Franek Mar 11 '15 at 13:25
  • And for $v(f\circ \phi)(p)$? Is it just emphasis? – EQJ Mar 11 '15 at 13:28
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    It is the application of the vector $v$ on the smooth function $f\circ\phi$. The $p$ is probably superfluous here (if $v\in T_p M$ is understood). – Peter Franek Mar 11 '15 at 13:30

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For any tangent-vector $v_p$ we have

$$v_p(f)=\text{df}\left(v_p\right)=\sum _{k=1}^n v^k\frac{\partial f}{\partial x^k}(p)$$

$v_p$ is mapped by $ T_p\phi$ into a tangent-vector, say

$$w_{\phi (p)}=T_p \phi \left(v_p\right)$$

Now

$$w_{\phi (p)}(g)=T_p \phi \left(v_p\right)(g)=v_p(g\circ \phi )$$

This is in local coordinates:

$$v_p(g\circ \phi )=\sum _{k=1}^n v^k\frac{\partial g\circ \phi }{\partial x^k}(p)$$

Applying chainrule this is

$$w_{\phi (p)}(g)=\sum _{l=1}^n \left(\sum _{k=1}^n v^k\frac{\partial x^l\circ \phi }{\partial x^k}(p)\right)\frac{\partial g}{\partial x^l}(\phi (p))$$

Therefore:

$$w_{\phi (p)}=\sum _{l=1}^n \left(\sum _{k=1}^n v^k\frac{\partial x^l\circ \phi }{\partial x^k}(p)\right)\frac{\partial }{\partial x^l}\phi (p)$$

The notation $d$ in do Carmo $\text{d$\phi $}$ for tangentmap is not a good one. $d$ means differential or exterior derivative. Upper indices are not powers!

Frieder
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