About the second fundamental form












12












$begingroup$



Let $Usubsetmathbb R^3$ be an open set, and $f:Uto mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}({0})$ is non-empty, and that at each $pin S,$ the gradient $overrightarrow nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $pmapsto overrightarrow eta(p)=frac{1}{||overrightarrow nabla f(p)||}overrightarrow nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $xin U,$ write $H(f)_{(x)}$ for the $3times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=frac{partial^2f}{partial x_ipartial x_j}(x).$$
Show that, at each $pin S$, the second fundamental form $II_p: T_p(s)times T_p(s)to mathbb R$ is the symmetric bilinear map
$$II_p(overrightarrow v,overrightarrow w)=frac{-1}{||overrightarrow nabla f(p)||}overrightarrow vcdot H(f)_{(p)}overrightarrow w,$$for all $overrightarrow v ,overrightarrow w in T_p(s)$.



(Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span{ {overrightarrow eta(p)}})^{bot}$ of $mathbb R^3$.



Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.



Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-<d N_p(v),v>$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)to T_p(S)$ is the differential of the Gauss map.




Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.



I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...



Looking forward to an understandable explaination. Thanks in advance.




Edit 2:Furthermore, show that, at each point $pin S$, the expression
$$phi_p(z)=detpmatrix{-H(f)_{(p)}-zI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}$$
(the underlying matrix here is $4times 4$) defines a second-degree polynomial whose roots $lambda_1$ and $lambda_2$ are $||overrightarrow nabla f(p)||k_1$ and $||overrightarrow nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.



Also, if a non-zero vector $pmatrix {overrightarrow v \c}$ lies in the kernel of the $4times 4$ matrix $$pmatrix{-H(f)_{(p)}-lambda_jI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0},$$
then $vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$.











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  • 1




    $begingroup$
    Isn't this proved in any standard book about differential geometry? I remember that this was quite straightforward from definitions if things are done properly. But maybe I am wrong. I just don't wanna go through this again, I hate differential geometry!
    $endgroup$
    – Patrick Da Silva
    Feb 21 '12 at 6:01
















12












$begingroup$



Let $Usubsetmathbb R^3$ be an open set, and $f:Uto mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}({0})$ is non-empty, and that at each $pin S,$ the gradient $overrightarrow nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $pmapsto overrightarrow eta(p)=frac{1}{||overrightarrow nabla f(p)||}overrightarrow nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $xin U,$ write $H(f)_{(x)}$ for the $3times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=frac{partial^2f}{partial x_ipartial x_j}(x).$$
Show that, at each $pin S$, the second fundamental form $II_p: T_p(s)times T_p(s)to mathbb R$ is the symmetric bilinear map
$$II_p(overrightarrow v,overrightarrow w)=frac{-1}{||overrightarrow nabla f(p)||}overrightarrow vcdot H(f)_{(p)}overrightarrow w,$$for all $overrightarrow v ,overrightarrow w in T_p(s)$.



(Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span{ {overrightarrow eta(p)}})^{bot}$ of $mathbb R^3$.



Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.



Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-<d N_p(v),v>$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)to T_p(S)$ is the differential of the Gauss map.




Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.



I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...



Looking forward to an understandable explaination. Thanks in advance.




Edit 2:Furthermore, show that, at each point $pin S$, the expression
$$phi_p(z)=detpmatrix{-H(f)_{(p)}-zI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}$$
(the underlying matrix here is $4times 4$) defines a second-degree polynomial whose roots $lambda_1$ and $lambda_2$ are $||overrightarrow nabla f(p)||k_1$ and $||overrightarrow nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.



Also, if a non-zero vector $pmatrix {overrightarrow v \c}$ lies in the kernel of the $4times 4$ matrix $$pmatrix{-H(f)_{(p)}-lambda_jI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0},$$
then $vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$.











share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    Isn't this proved in any standard book about differential geometry? I remember that this was quite straightforward from definitions if things are done properly. But maybe I am wrong. I just don't wanna go through this again, I hate differential geometry!
    $endgroup$
    – Patrick Da Silva
    Feb 21 '12 at 6:01














12












12








12


4



$begingroup$



Let $Usubsetmathbb R^3$ be an open set, and $f:Uto mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}({0})$ is non-empty, and that at each $pin S,$ the gradient $overrightarrow nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $pmapsto overrightarrow eta(p)=frac{1}{||overrightarrow nabla f(p)||}overrightarrow nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $xin U,$ write $H(f)_{(x)}$ for the $3times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=frac{partial^2f}{partial x_ipartial x_j}(x).$$
Show that, at each $pin S$, the second fundamental form $II_p: T_p(s)times T_p(s)to mathbb R$ is the symmetric bilinear map
$$II_p(overrightarrow v,overrightarrow w)=frac{-1}{||overrightarrow nabla f(p)||}overrightarrow vcdot H(f)_{(p)}overrightarrow w,$$for all $overrightarrow v ,overrightarrow w in T_p(s)$.



(Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span{ {overrightarrow eta(p)}})^{bot}$ of $mathbb R^3$.



Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.



Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-<d N_p(v),v>$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)to T_p(S)$ is the differential of the Gauss map.




Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.



I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...



Looking forward to an understandable explaination. Thanks in advance.




Edit 2:Furthermore, show that, at each point $pin S$, the expression
$$phi_p(z)=detpmatrix{-H(f)_{(p)}-zI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}$$
(the underlying matrix here is $4times 4$) defines a second-degree polynomial whose roots $lambda_1$ and $lambda_2$ are $||overrightarrow nabla f(p)||k_1$ and $||overrightarrow nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.



Also, if a non-zero vector $pmatrix {overrightarrow v \c}$ lies in the kernel of the $4times 4$ matrix $$pmatrix{-H(f)_{(p)}-lambda_jI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0},$$
then $vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$.











share|cite|improve this question











$endgroup$





Let $Usubsetmathbb R^3$ be an open set, and $f:Uto mathbb R$ be a smooth function. Suppose that the level set $S=f^{-1}({0})$ is non-empty, and that at each $pin S,$ the gradient $overrightarrow nabla f(p)$ is not the zero vector. Then $S$ is a smooth two-dimensional surface in $U$, and $pmapsto overrightarrow eta(p)=frac{1}{||overrightarrow nabla f(p)||}overrightarrow nabla f(p)$ defines a smooth unit-length normal vector field along $S$. At each $xin U,$ write $H(f)_{(x)}$ for the $3times 3$ Hessian matrix specified by $$(H(f)_{(x)})_{ij}=frac{partial^2f}{partial x_ipartial x_j}(x).$$
Show that, at each $pin S$, the second fundamental form $II_p: T_p(s)times T_p(s)to mathbb R$ is the symmetric bilinear map
$$II_p(overrightarrow v,overrightarrow w)=frac{-1}{||overrightarrow nabla f(p)||}overrightarrow vcdot H(f)_{(p)}overrightarrow w,$$for all $overrightarrow v ,overrightarrow w in T_p(s)$.



(Here, we view the tagent space $T_p(S)$ as the two-dimensional subspace $(span{ {overrightarrow eta(p)}})^{bot}$ of $mathbb R^3$.



Edit: Actually my question is why the second fundamental form under the usual definition can be written in this way.



Definition: The quadratic form $II_p$, defined in $T_p(S)$ by $II_p(v)=-<d N_p(v),v>$ is called the second fundamental form of $S$ at $p$, where $dN_p:T_p(S)to T_p(S)$ is the differential of the Gauss map.




Hopefully, I express this problem explicitly. I was just wondering how to prove this statement.



I took a diffrential geometry class last semester, and when I organized my notes this morning, I found this statement, but there was no proof...



Looking forward to an understandable explaination. Thanks in advance.




Edit 2:Furthermore, show that, at each point $pin S$, the expression
$$phi_p(z)=detpmatrix{-H(f)_{(p)}-zI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}$$
(the underlying matrix here is $4times 4$) defines a second-degree polynomial whose roots $lambda_1$ and $lambda_2$ are $||overrightarrow nabla f(p)||k_1$ and $||overrightarrow nabla f(p)||k_2$, where $k_1$ and $k_2$ are the principal curvatures of $S$ at $p$.



Also, if a non-zero vector $pmatrix {overrightarrow v \c}$ lies in the kernel of the $4times 4$ matrix $$pmatrix{-H(f)_{(p)}-lambda_jI_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0},$$
then $vec v$ is a non-zero element of $T_p(S)$ and lies in the "principal direction" corresponding to $K_j$.








differential-geometry






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edited Feb 21 '12 at 18:32









Ehsan M. Kermani

6,43412448




6,43412448










asked Feb 21 '12 at 5:10









PetersonPeterson

636




636








  • 1




    $begingroup$
    Isn't this proved in any standard book about differential geometry? I remember that this was quite straightforward from definitions if things are done properly. But maybe I am wrong. I just don't wanna go through this again, I hate differential geometry!
    $endgroup$
    – Patrick Da Silva
    Feb 21 '12 at 6:01














  • 1




    $begingroup$
    Isn't this proved in any standard book about differential geometry? I remember that this was quite straightforward from definitions if things are done properly. But maybe I am wrong. I just don't wanna go through this again, I hate differential geometry!
    $endgroup$
    – Patrick Da Silva
    Feb 21 '12 at 6:01








1




1




$begingroup$
Isn't this proved in any standard book about differential geometry? I remember that this was quite straightforward from definitions if things are done properly. But maybe I am wrong. I just don't wanna go through this again, I hate differential geometry!
$endgroup$
– Patrick Da Silva
Feb 21 '12 at 6:01




$begingroup$
Isn't this proved in any standard book about differential geometry? I remember that this was quite straightforward from definitions if things are done properly. But maybe I am wrong. I just don't wanna go through this again, I hate differential geometry!
$endgroup$
– Patrick Da Silva
Feb 21 '12 at 6:01










1 Answer
1






active

oldest

votes


















6












$begingroup$

I will use the notation $df_p$ rather than $vec{nabla}f(p)$.



First, we differentiate the function $N : p mapsto frac{df_p}{|df_p|}$ and get



$$dN_p(v) = frac{H(f)_pv}{|df_p|} - frac{df_p cdot H(f)_pv}{|df_p|^3} df_p .$$



Therefore



$$langle dN_p(v),w rangle = frac{left
((df_p cdot df_p) w - (df_pcdot w)df_pright)H(f)_pv}{|df_p|^3} \ = frac{left
(df_p wedge(w wedge df_p)right)H(f)_pv}{|df_p|^3} = frac{w H(f)_p v}{|df_p|}.$$



For your second question, first of all it is not difficult to see that $phi_p$ is a second-degree polynomial.



Notice that by properties of second fundamental form, $|df_p|k_1$ and $|df_p|k_2$ are eigenvalues of $-H(f)_p$ with eigenvectors "principal direction" corresponding to $k_1$ and $k_2$ respectively, denoted $v_1$ and $v_2$. Thus it is clear that the vectors $(v_i,0)$ lie in the kernel of



$$pmatrix{-H(f)_{(p)}-|df_p|k_i I_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}.$$



Hence $phi_p(|df_p|k_i) = 0$ for $i= 1,2$.





EDIT (to clarify the two points mentioned in your comments):





  1. $phi_p$ is at most a third-degree polynomial. If we develop the determinant, the summand of $z^3$ is necessary the product of diagonal coefficients, which is zero.

  2. Firstly, $v_i$ is in the kernel of $-H(f)_{p}-|df_p|k_i I_{3times 3}$. Since $v_i$ lies in the tangent space and $df_p$ is a normal vector, one also has $df_pcdot v_i = 0$. So the product of this $4times 4$ matrix by $(v_i,0)$ is zero.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51












  • $begingroup$
    Could you explain more about them?
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51










  • $begingroup$
    I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
    $endgroup$
    – Jacob Ikabruob
    Feb 21 '12 at 20:11












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1 Answer
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1 Answer
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active

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active

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active

oldest

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6












$begingroup$

I will use the notation $df_p$ rather than $vec{nabla}f(p)$.



First, we differentiate the function $N : p mapsto frac{df_p}{|df_p|}$ and get



$$dN_p(v) = frac{H(f)_pv}{|df_p|} - frac{df_p cdot H(f)_pv}{|df_p|^3} df_p .$$



Therefore



$$langle dN_p(v),w rangle = frac{left
((df_p cdot df_p) w - (df_pcdot w)df_pright)H(f)_pv}{|df_p|^3} \ = frac{left
(df_p wedge(w wedge df_p)right)H(f)_pv}{|df_p|^3} = frac{w H(f)_p v}{|df_p|}.$$



For your second question, first of all it is not difficult to see that $phi_p$ is a second-degree polynomial.



Notice that by properties of second fundamental form, $|df_p|k_1$ and $|df_p|k_2$ are eigenvalues of $-H(f)_p$ with eigenvectors "principal direction" corresponding to $k_1$ and $k_2$ respectively, denoted $v_1$ and $v_2$. Thus it is clear that the vectors $(v_i,0)$ lie in the kernel of



$$pmatrix{-H(f)_{(p)}-|df_p|k_i I_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}.$$



Hence $phi_p(|df_p|k_i) = 0$ for $i= 1,2$.





EDIT (to clarify the two points mentioned in your comments):





  1. $phi_p$ is at most a third-degree polynomial. If we develop the determinant, the summand of $z^3$ is necessary the product of diagonal coefficients, which is zero.

  2. Firstly, $v_i$ is in the kernel of $-H(f)_{p}-|df_p|k_i I_{3times 3}$. Since $v_i$ lies in the tangent space and $df_p$ is a normal vector, one also has $df_pcdot v_i = 0$. So the product of this $4times 4$ matrix by $(v_i,0)$ is zero.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51












  • $begingroup$
    Could you explain more about them?
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51










  • $begingroup$
    I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
    $endgroup$
    – Jacob Ikabruob
    Feb 21 '12 at 20:11
















6












$begingroup$

I will use the notation $df_p$ rather than $vec{nabla}f(p)$.



First, we differentiate the function $N : p mapsto frac{df_p}{|df_p|}$ and get



$$dN_p(v) = frac{H(f)_pv}{|df_p|} - frac{df_p cdot H(f)_pv}{|df_p|^3} df_p .$$



Therefore



$$langle dN_p(v),w rangle = frac{left
((df_p cdot df_p) w - (df_pcdot w)df_pright)H(f)_pv}{|df_p|^3} \ = frac{left
(df_p wedge(w wedge df_p)right)H(f)_pv}{|df_p|^3} = frac{w H(f)_p v}{|df_p|}.$$



For your second question, first of all it is not difficult to see that $phi_p$ is a second-degree polynomial.



Notice that by properties of second fundamental form, $|df_p|k_1$ and $|df_p|k_2$ are eigenvalues of $-H(f)_p$ with eigenvectors "principal direction" corresponding to $k_1$ and $k_2$ respectively, denoted $v_1$ and $v_2$. Thus it is clear that the vectors $(v_i,0)$ lie in the kernel of



$$pmatrix{-H(f)_{(p)}-|df_p|k_i I_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}.$$



Hence $phi_p(|df_p|k_i) = 0$ for $i= 1,2$.





EDIT (to clarify the two points mentioned in your comments):





  1. $phi_p$ is at most a third-degree polynomial. If we develop the determinant, the summand of $z^3$ is necessary the product of diagonal coefficients, which is zero.

  2. Firstly, $v_i$ is in the kernel of $-H(f)_{p}-|df_p|k_i I_{3times 3}$. Since $v_i$ lies in the tangent space and $df_p$ is a normal vector, one also has $df_pcdot v_i = 0$. So the product of this $4times 4$ matrix by $(v_i,0)$ is zero.






share|cite|improve this answer











$endgroup$













  • $begingroup$
    Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51












  • $begingroup$
    Could you explain more about them?
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51










  • $begingroup$
    I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
    $endgroup$
    – Jacob Ikabruob
    Feb 21 '12 at 20:11














6












6








6





$begingroup$

I will use the notation $df_p$ rather than $vec{nabla}f(p)$.



First, we differentiate the function $N : p mapsto frac{df_p}{|df_p|}$ and get



$$dN_p(v) = frac{H(f)_pv}{|df_p|} - frac{df_p cdot H(f)_pv}{|df_p|^3} df_p .$$



Therefore



$$langle dN_p(v),w rangle = frac{left
((df_p cdot df_p) w - (df_pcdot w)df_pright)H(f)_pv}{|df_p|^3} \ = frac{left
(df_p wedge(w wedge df_p)right)H(f)_pv}{|df_p|^3} = frac{w H(f)_p v}{|df_p|}.$$



For your second question, first of all it is not difficult to see that $phi_p$ is a second-degree polynomial.



Notice that by properties of second fundamental form, $|df_p|k_1$ and $|df_p|k_2$ are eigenvalues of $-H(f)_p$ with eigenvectors "principal direction" corresponding to $k_1$ and $k_2$ respectively, denoted $v_1$ and $v_2$. Thus it is clear that the vectors $(v_i,0)$ lie in the kernel of



$$pmatrix{-H(f)_{(p)}-|df_p|k_i I_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}.$$



Hence $phi_p(|df_p|k_i) = 0$ for $i= 1,2$.





EDIT (to clarify the two points mentioned in your comments):





  1. $phi_p$ is at most a third-degree polynomial. If we develop the determinant, the summand of $z^3$ is necessary the product of diagonal coefficients, which is zero.

  2. Firstly, $v_i$ is in the kernel of $-H(f)_{p}-|df_p|k_i I_{3times 3}$. Since $v_i$ lies in the tangent space and $df_p$ is a normal vector, one also has $df_pcdot v_i = 0$. So the product of this $4times 4$ matrix by $(v_i,0)$ is zero.






share|cite|improve this answer











$endgroup$



I will use the notation $df_p$ rather than $vec{nabla}f(p)$.



First, we differentiate the function $N : p mapsto frac{df_p}{|df_p|}$ and get



$$dN_p(v) = frac{H(f)_pv}{|df_p|} - frac{df_p cdot H(f)_pv}{|df_p|^3} df_p .$$



Therefore



$$langle dN_p(v),w rangle = frac{left
((df_p cdot df_p) w - (df_pcdot w)df_pright)H(f)_pv}{|df_p|^3} \ = frac{left
(df_p wedge(w wedge df_p)right)H(f)_pv}{|df_p|^3} = frac{w H(f)_p v}{|df_p|}.$$



For your second question, first of all it is not difficult to see that $phi_p$ is a second-degree polynomial.



Notice that by properties of second fundamental form, $|df_p|k_1$ and $|df_p|k_2$ are eigenvalues of $-H(f)_p$ with eigenvectors "principal direction" corresponding to $k_1$ and $k_2$ respectively, denoted $v_1$ and $v_2$. Thus it is clear that the vectors $(v_i,0)$ lie in the kernel of



$$pmatrix{-H(f)_{(p)}-|df_p|k_i I_{3times 3} & overrightarrow nabla f(p)\ pm overrightarrow nabla f(p)& 0}.$$



Hence $phi_p(|df_p|k_i) = 0$ for $i= 1,2$.





EDIT (to clarify the two points mentioned in your comments):





  1. $phi_p$ is at most a third-degree polynomial. If we develop the determinant, the summand of $z^3$ is necessary the product of diagonal coefficients, which is zero.

  2. Firstly, $v_i$ is in the kernel of $-H(f)_{p}-|df_p|k_i I_{3times 3}$. Since $v_i$ lies in the tangent space and $df_p$ is a normal vector, one also has $df_pcdot v_i = 0$. So the product of this $4times 4$ matrix by $(v_i,0)$ is zero.







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Jan 7 at 19:34









Henrik Schumacher

35527




35527










answered Feb 21 '12 at 19:41









Jacob IkabruobJacob Ikabruob

492210




492210












  • $begingroup$
    Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51












  • $begingroup$
    Could you explain more about them?
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51










  • $begingroup$
    I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
    $endgroup$
    – Jacob Ikabruob
    Feb 21 '12 at 20:11


















  • $begingroup$
    Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51












  • $begingroup$
    Could you explain more about them?
    $endgroup$
    – Peterson
    Feb 21 '12 at 19:51










  • $begingroup$
    I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
    $endgroup$
    – Jacob Ikabruob
    Feb 21 '12 at 20:11
















$begingroup$
Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
$endgroup$
– Peterson
Feb 21 '12 at 19:51






$begingroup$
Thanks, about the second question, I was wondering why "it is not difficult to see that $phi_p$ is a second-degree polynomial" and "Thus it is clear that the vectors $(v_i,c)$ lies in the kernel".
$endgroup$
– Peterson
Feb 21 '12 at 19:51














$begingroup$
Could you explain more about them?
$endgroup$
– Peterson
Feb 21 '12 at 19:51




$begingroup$
Could you explain more about them?
$endgroup$
– Peterson
Feb 21 '12 at 19:51












$begingroup$
I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
$endgroup$
– Jacob Ikabruob
Feb 21 '12 at 20:11




$begingroup$
I've just edited of my post to make some clarifications. By the way, I don't think $(v_i,c)$ lies in the kernel for all $c$. Only $(v_i,0)$ does.
$endgroup$
– Jacob Ikabruob
Feb 21 '12 at 20:11


















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