
Equation:  Chapter .. 1 




The .. principal .. curvat ures  . 





 1   .   1  .. Volume .. of a thickened .. hypersurface

 
We want to consider the following problem  :  Let  Y   subset   R  to the power of  n  be an oriented hyper  hyphen 





surface, so there is a well defined unit normal vector,  nu   open parenthesis   y   closing parenthesis   ,  at each point of  Y   . 





Let  Y  sub  h  denote the set of all points of the form




 y   plus   t   nu   open parenthesis   y   closing parenthesis   ,   0   less or equal    t   less or equal    h.  




We wish to compute  V  sub  n   open parenthesis   Y  sub  h   closing parenthesis  where  V  sub  n  denotes the  n   minus  dimensional volume. We





will do this computation for small  h   ,  see the discussion after the examples.





Examples in three dimensional space.





 1   .  Suppose that  Y  is a bounded region in a plane, of area  A   .  Clearly




 V  sub  3   open parenthesis   Y  sub  h   closing parenthesis   =   hA 




in this case.





 2   .  .. Suppose that  Y  is a right circular cylinder of radius  r  and height  l  with

 
outwardly pointing normal. .. Then  Y  sub  h  is the region between the right circular





cylinders of height  l  and radii  r  and  r   plus   h  so




Line 1  V  sub  3   open parenthesis   Y  sub  h   closing parenthesis   =   pi   l   open square bracket   open parenthesis   r   plus   h   closing parenthesis  to the power of  2   minus   r  to the power of  2   closing square bracket  Line 2  =   2   pi   l   rh   plus   pi   l   h  to the power of  2  Line 3  =   hA   plus   h  to the power of  2   times   1  divided by  2   r   times   A  Line 4  =   A   parenleftbigg   h   plus   1  divided by  2   times   kh  to the power of  2   parenrightbigg   , 




where  A   =   2   pi   r   l  is the area of the cylinder and where  k   =   1   slash   r  is the curvature of

 
the generating circle of the cylinder. For small  h   ,  this formula is correct, in fact,




 11 




 12  .. CHAPTER  1   .  .. THE PRINCIPAL CURVATURES.

 
whether we choose the normal vector to point out of the cylinder or into the





cylinder. Of course, in the inward pointing case, the curvature has the opposite




 sign,    k   =   minus   1   slash   r   . 




For inward pointing normals, the formula breaks down when  h   >   r   ,  since we





get multiple coverage of points in space by points of the form  y   plus   t   nu   open parenthesis   y   closing parenthesis   . 





 3   .   Y  is a sphere of radius  R  with outward normal, so  Y  sub  h  is a spherical shell,

 
and




Line 1  V  sub  3   open parenthesis   Y  sub  h   closing parenthesis   =   4  divided by  3   pi   open square bracket   open parenthesis   R   plus   h   closing parenthesis  to the power of  3   minus   R  to the power of  3   closing square bracket  Line 2  =   h   4   pi   R  to the power of  2   plus   h  to the power of  2   4   pi   R   plus   h  to the power of  3   4  divided by  3   pi  Line 3  =   hA   plus   h  to the power of  2   1  divided by  R   A   plus   h  to the power of  3   1  divided by  3   R  to the power of  2   A  Line 4  =   1  divided by  3   times   A   times   bracketleftbigg   3   h   plus   3   1  divided by  R   times   h  to the power of  2   plus   1  divided by  R  to the power of  2   h  to the power of  3   bracketrightbigg   , 




where  A   =   4   pi   R  to the power of  2  is the area of the sphere.





Once again, for inward pointing normals we must change the sign of the





coefficient of  h  to the power of  2  and the formula thus obtained is only correct for  h   less or equal    1  divided by  R   . 





So in general, we wish to make the assumption that  h  is such that the map




 Y   times   open square bracket   0   ,   h   closing square bracket   right arrow   R  to the power of  n   ,   open parenthesis   y,    t   closing parenthesis   mapsto-arrowright   y   plus   t   nu   open parenthesis   y   closing parenthesis 




is injective. .. For  Y  compact, .. there always exists an  h  sub  0   >   0  such that this

 
condition holds for all  h   less   h  sub  0   .  .. This can be seen to be a consequence of the

 
implicit function theorem. .. But so not to interrupt the discussion, we will take

 
the injectivity of the map as an hypothesis, for the moment.





In a moment we will define the notion of the various averaged curvatures,

 
 H  sub  1   ,   .   .   .   ,   H  sub  n   minus   1   ,  ofa hypersurface, and find for the case ofthe sphere with outward

 
pointing normal, that




 H  sub  1   =   1  divided by  R   ,   H  sub  2   =   1  divided by  R  to the power of  2   , 




while for the case of the cylinder with outward pointing normal that




 H  sub  1   =   1  divided by  2   r   ,   H  sub  2   =   0   , 




and for the case of the planar region that




 H  sub  1   =   H  sub  2   =   0   . 




We can thus write all three of the above the above formulas as




 V  sub  3   open parenthesis   Y  sub  h   closing parenthesis   =   1  divided by  3   A   bracketleftbig   3   h   plus   3   H  sub  1   h  to the power of  2   plus   H  sub  2   h  to the power of  3   bracketrightbig   . 




 1   .   2   .  .... THE GAUSS MAP AND THE WEINGARTEN MAP. ....  13 





 1   .   2  .. The .. Gauss .. map .. and .. the .. Weingarten .. map.

 
In order to state the general formula, we make the following definitions  :  .. Let





 Y  be an  open parenthesis  immersed  closing parenthesis  oriented hypersurface. .... At each  x   in   Y  there is a unique





 open parenthesis  positive  closing parenthesis  unit normal vector, and hence a well defined Gauss map




 nu   :   Y   right arrow   S  to the power of  n   minus   1 




assigning to each point  x   in   Y  its unit normal vector,  nu   open parenthesis   x   closing parenthesis   .  .. Here  S  to the power of  n   minus   1  denotes

 
the unit sphere, the set of all unit vectors in  R  to the power of  n   . 





The normal vector,  nu   open parenthesis   x   closing parenthesis  is orthogonal to the tangent space to  Y  at  x   .  .. We

 
will denote this tangent space by  TY  sub  x   .  For our present purposes, we can regard

 
 TY  sub  x  as a subspace of  R  to the power of  n   :  .. If  t   right arrow   gamma   open parenthesis   t   closing parenthesis  is a di  ff  erentiable curve lying on the

 
hypersurface  Y   ,   open parenthesis  this means that  gamma   open parenthesis   t   closing parenthesis   in   Y  for all  t   closing parenthesis  and if  gamma   open parenthesis   0   closing parenthesis   =   x   ,  then  gamma  to the power of  prime   open parenthesis   0   closing parenthesis 

 
belongs to the tangent space  TY  sub  x   .  .. Conversely, given any vector  v   in   TY  sub  x   ,  we

 
can always find a di  ff  erentiable curve  gamma  with  gamma   open parenthesis   0   closing parenthesis   =   x,    gamma  to the power of  prime   open parenthesis   0   closing parenthesis   =   v   .  .. So a good

 
way to think of a tangent vector to  Y  at  x  is as an  quotedblleft  infinitesimal curve  quotedblright  on  Y 

 
passing through  x   . 




 Examples   : 




 1   .  .. Suppose that  Y  is a portion of an  open parenthesis   n   minus   1   closing parenthesis  dimensional linear or affine sub  hyphen 





space space sitting in  R  to the power of  n   .  For example suppose that  Y   =   R  to the power of  n   minus   1  consisting

 
of those points in  R  to the power of  n  whose last coordinate vanishes. .. Then the tangent

 
space to  Y  at every point is just this same subspace, and hence the normal

 
vector is a constant. The Gauss map is thus a constant, mapping all of  Y 





onto a single point in  S  to the power of  n   minus   1   . 





 2   .  .. Suppose that  Y  is the sphere of radius  R   open parenthesis  say centered at the origin  closing parenthesis   .  The





Gauss map carries every point of  Y  into the corresponding  open parenthesis  parallel  closing parenthesis  point





of  S  to the power of  n   minus   1   .  In other words, it is multiplication by  1   slash   R   : 




 nu   open parenthesis   y   closing parenthesis   =   1  divided by  R   y.  




 3   .  .. Suppose that  Y  is a right circular cylinder in  R  to the power of  3  whose base is the circle

 
of radius  r  in the  x  to the power of  1   ,   x  to the power of  2  plane. .. Then the Gauss map sends  Y  onto the

 
equator of the unit sphere,  S  to the power of  2   ,  sending a point  x  into  open parenthesis   1   slash   r   closing parenthesis   pi   open parenthesis   x   closing parenthesis  where





 pi   :   R  to the power of  3   right arrow   R  to the power of  2  is projection onto the  x  to the power of  1   ,   x  to the power of  2  plane.





Another good way to think of the tangent space is in terms of a local





parameterization which means that we are given a map  X   :   M   right arrow   R  to the power of  n  where

 
 M  is some open subset of  R  to the power of  n   minus   1  and such that  X   open parenthesis   M   closing parenthesis  is some neighborhood of





 x  in  Y   .  .... Let  y  to the power of  1   ,   .   .   .   ,   y  to the power of  n   minus   1  be the standard coordinates on  R  to the power of  n   minus   1   .  .... Part of the





requirement that goes into the definition of parameterization is that the map  X 





be regular  ,  in the sense that its Jacobian matrix




 dX   :   =   parenleftbigg   partialdiff   X  divided by  partialdiff   y  to the power of  1   ,   times   times   times   ,   partialdiff   X  divided by  partialdiff   y  to the power of  n   minus   1   parenrightbigg 




 14  .... CHAPTER  1   .  .... THE PRINCIPAL CURVATURES.





whose columns are the partial derivatives of the map  X  has rank  n   minus   1  every  hyphen 

 
where. The matrix  dX  has  n  rows and  n   minus   1  columns. The regularity condition

 
amounts to the assertion that for each  z   in   M  the vectors,




 partialdiff   X  divided by  partialdiff   y  to the power of  1   open parenthesis   z   closing parenthesis   ,   times   times   times   ,   partialdiff   X  divided by  partialdiff   y  to the power of  n   minus   1   open parenthesis   z   closing parenthesis 




span a subspace of dimension  n   minus   1   .  If  x   =   X   open parenthesis   y   closing parenthesis  then the tangent space  TY  sub  x  is

 
precisely the space spanned by




 partialdiff   X  divided by  partialdiff   y  to the power of  1   open parenthesis   y   closing parenthesis   ,   times   times   times   ,   partialdiff   X  divided by  partialdiff   y  to the power of  n   minus   1   open parenthesis   y   closing parenthesis   . 




Suppose that  F  is a di  ff  erentiable map from  Y  to  R  to the power of  m   .  .. We can then define

 
its di  ff  erential,  dF  sub  x   :   TY  sub  x   arrowright-mapsto   R  to the power of  m   .  .. It is a linear map assigning to each  v   in   TY  sub  x 

 
a value  dF  sub  x   open parenthesis   v   closing parenthesis   in   R  to the power of  m   :  .. In terms of the ..  quotedblleft  infinitesimal curve  quotedblright  .. description, .. if




 v   =   gamma  to the power of  prime   open parenthesis   0   closing parenthesis   then 
 dF  sub  x   open parenthesis   v   closing parenthesis   =   dF   circ   gamma  divided by  dt   open parenthesis   0   closing parenthesis   . 




 open parenthesis  You must check that this does not depend on the choice of representing curve,




 gamma   .   closing parenthesis 




Alternatively, .. to give a linear map, .. it is enough to give its value at the

 
elements of a basis. .. In terms of the basis coming from a parameterization, we

 
have




 dF  sub  x   parenleftbigg   partialdiff   X  divided by  partialdiff   y  to the power of  i   open parenthesis   y   closing parenthesis   parenrightbigg   =   partialdiff   F   circ   X  divided by  partialdiff   y  to the power of  i   open parenthesis   y   closing parenthesis   . 




Here  F   circ   X   :   M   right arrow   R  to the power of  m  is the composition of the map  F  with the map  X   .  You

 
must check that the map  dF  sub  x  so determined does not depend on the choice of

 
parameterization. Both of these verifications proceed by the chain rule.





One immediate consequence of either characterization is the following im  hyphen 

 
portant property. .. Suppose that  F  takes values in a submanifold  Z   subset   R  to the power of  m   . 

 
Then




 dF  sub  x   :   TY  sub  x   right arrow   TZ  sub  F   open parenthesis   x   closing parenthesis   . 




Let us apply all this to the Gauss map,  nu   ,  which maps  Y  to the unit sphere,




 S  to the power of  n   minus   1   .   Then 
 d   nu  sub  x   :   TY  sub  x   right arrow   TS   n   nu   minus   open parenthesis   x   1 closing parenthesis  . 




But the tangent space to the unit sphere at  nu   open parenthesis   x   closing parenthesis  .. consists of all vectors

 
perpendicular to  nu   open parenthesis   x   closing parenthesis  and so can be identified with  TY  sub  x   .  We define the Wein  hyphen 

 
garten map to be the di  ff  erential of the Gauss map, regarded as a map from

 
 TY  sub  x  to itself  : 




 W  sub  x   :   =   d   nu  sub  x   ,   W  sub  x   :   TY  sub  x   right arrow   TY  sub  x   . 




The second fundamental form is defined to be the bilinear form on  TY  sub  x 

 
given by




 II  sub  x   open parenthesis   v,    w   closing parenthesis   :   =   open parenthesis   W  sub  x   v,    w   closing parenthesis   . 




 1   .   2   .  .. THE GAUSS MAP AND THE WEINGARTEN MAP. ..  15 

 
In the next section we will show, .. using local coordinates, .. that this form is





symmetric, i.e. that




 open parenthesis   W  sub  x   u,    v   closing parenthesis   =   open parenthesis   u,    W  sub  x   v   closing parenthesis   . 




This implies, from linear algebra, that  W  sub  x  is diagonizable with real eigenvalues.

 
These eigenvalues,  k  sub  1   =   k  sub  1   open parenthesis   x   closing parenthesis   ,   times   times   times   ,   k  sub  n   minus   1   =   k  sub  n   minus   1   open parenthesis   x   closing parenthesis   ,  of the Weingarten map are

 
called the principal curvatures of  Y  at the point  x   . 




 Examples   : 




 1   .  .. For a portion of  open parenthesis   n   minus   1   closing parenthesis  space sitting in  R  to the power of  n  the Gauss map is constant

 
so its di  ff  erential is zero. .. Hence the Weingarten map and thus all the

 
principal curvatures are zero.





 2   .  .. For the sphere ofradius  R  the Gauss map consists ofmultiplication by  1   slash   R 

 
which is a linear transformation. The di  ff  erential ofa linear transformation





is that same transformation  open parenthesis  regarded as acting on the tangent spaces  closing parenthesis   . 

 
Hence the Weingarten map is  1   slash   R   times  id and so all the principal curvatures

 
are equal and are equal to  1   slash   R   . 





 3   .  .. For the cylinder, .. again the Gauss map is linear, .. and so the principal

 
curvatures are  0  and  1   slash   r   . 





We let  H  sub  j  denote the  j  th normalized elementary symmetric functions of the

 
principal curvatures. So




Line 1  H  sub  0   =   1  Line 2  H  sub  1   =   1  divided by  n   minus   1   open parenthesis   k  sub  1   plus   times   times   times   plus   k  sub  n   minus   1   closing parenthesis  Line 3  H  sub  n   minus   1   =   k  sub  1   times   k  sub  2   times   times   times   k  sub  n   minus   1 




and, in general,




Equation:  H  sub  j   =   parenleftbigg   n   j   1   parenrightbigg  sub  minus   1   1   less or equal    i   1   less   times   sum   times   times   less   i   j   less or equal    n   minus   1   k  sub  i   1   times   times   times   k  sub  i   j   . .. open parenthesis   1   .   1   closing parenthesis 




 H  sub  1  is called the mean curvature and  H  sub  n   minus   1  is called the Gaussian curvature  . 

 
All the principal curvatures are functions of the point  x   in   Y   .  .. For notational

 
simplicity, we will frequently suppress the dependence on  x   .  .. Then the formula

 
for the volume of the thickened hypersurface ..  open parenthesis  we will call this the ..  quotedblleft  volume





formula  quotedblright  for short  closing parenthesis  is  : 




Equation:  V  sub  n   open parenthesis   Y  sub  h   closing parenthesis   =   1  divided by  n   sum  from  i   =   1  to  n  Row 1  n  Row 2  i .  h  to the power of  i   integral  sub  Y   H  sub  i   minus   1   d  to the power of  n   minus   1   A .. open parenthesis   1   .   2   closing parenthesis 




where  d  to the power of  n   minus   1   A  denotes the  open parenthesis   n   minus   1  dimensional  closing parenthesis  volume  open parenthesis  area  closing parenthesis  measure on  Y   . 





A immediate check shows that this gives the answers that we got above for

 
the the plane, the cylinder, and the sphere.





 16  .... CHAPTER  1   .  .... THE PRINCIPAL CURVATURES.





 1   .   3  .. Proof of the .. volume .. formula.





We recall that the Gauss map,  nu  assigns to each point  x   in   Y  its unit normal





vector, and so is a map from  Y  to the unit sphere,  S  to the power of  n   minus   1   .  The Weingarten map,

 
 W  sub  x   ,  is the di  ff  erential of the Gauss map,  W  sub  x   =   d   nu  sub  x   ,  regarded as a map of the

 
tangent space,  TY  sub  x  to itself. .. We now describe these maps in terms of a local

 
parameterization of  Y   .  .. So let  X   :   M   right arrow   R  to the power of  n  be a parameterization of class





 C  to the power of  2  of a neighborhood of  Y  near  x   ,  where  M  is an open subset of  R  to the power of  n   minus   1   .  .... So




Equation:  x   =   X   open parenthesis   y   closing parenthesis   , .. y   in   M   ,   say.    Let 
 N   :   nu   X 




so that  N   :   M   right arrow   S  to the power of  n   minus   1  is a map of class  C  to the power of  1   .  The map




 dX  sub  y   :   R  to the power of  n   minus   1   right arrow   TY  sub  x 




gives a frame of  TY  sub  x   .  The word  quotedblleft  frame  quotedblright  means an isomorphism of our  quotedblleft  stan  hyphen 





dard  quotedblright   open parenthesis   n   minus   1   closing parenthesis   hyphen  dimensional space,  R  to the power of  n   minus   1  with our given  open parenthesis   n   minus   1   closing parenthesis   hyphen  dimensional space,





 TY  sub  x   .  Here we have identified  T   open parenthesis   R  to the power of  n   minus   1   closing parenthesis  sub  y  with  R  to the power of  n   minus   1   ,  so the frame  dX  sub  y  gives us





a particular isomorphism of  R  to the power of  n   minus   1  with  TY  sub  x   . 





Giving a frame of a vector space is the same as giving a basis of that vector

 
space. .. We will use these two di  ff  erent ways of using the word  quotedblleft  frame  quotedblright  .. inter  hyphen 





changeably. .. Let  e  sub  1   ,   .   .   .   ,   e  sub  n   minus   1  .. denote the standard basis of  R  to the power of  n   minus   1   ,  .. and for  X 

 
and  N   ,  let the subscript  i  denote the partial derivative with respect to the  i  th

 
Cartesian coordinate. Thus




 dX  sub  y   open parenthesis   e  sub  i   closing parenthesis   =   X  sub  i   open parenthesis   y   closing parenthesis 




for example, .... and so  X  sub  1   open parenthesis   y   closing parenthesis   ,   .   .   .   ,   X  sub  n   minus   1   open parenthesis   y   closing parenthesis   quotedblleft  is  quotedblright  .... the frame determined by  dX  sub  y 





 open parenthesis  when we regard  TY  sub  x  as a subspace of  R  to the power of  n   closing parenthesis   .  .. For the sake of notational sim  hyphen 

 
plicity we will drop the argument  y   .  Thus we have




Line 1  dX   open parenthesis   e  sub  i   closing parenthesis   =   X  sub  i   ,  Line 2  dN   open parenthesis   e  sub  i   closing parenthesis   =   N  sub  i   , 




and so




 W  sub  x   X  sub  i   =   N  sub  i   . 




Recall the definition,  II  sub  x   open parenthesis   v,    w   closing parenthesis   =   open parenthesis   W  sub  x   v,    w   closing parenthesis   ,  of the second fundamental form.

 
Let  open parenthesis   L  sub  ij   closing parenthesis  denote the matrix of the second fundamental form with respect to the

 
basis  X  sub  1   ,   .   .   .   X  sub  n   minus   1  of  TY  sub  x   .  So




Line 1  L  sub  ij   =   II  sub  x   open parenthesis   X  sub  i   ,   X  sub  j   closing parenthesis  Line 2  =   open parenthesis   W  sub  x   X  sub  i   ,   X  sub  j   closing parenthesis  Line 3  =   open parenthesis   N  sub  i   ,   X  sub  j   closing parenthesis 




so




Equation:  L  sub  ij   =   minus   open parenthesis   N,    partialdiff  to the power of  2   X  divided by  partialdiff   y  sub  i   partialdiff   y  sub  j   closing parenthesis   , .. open parenthesis   1   .   3   closing parenthesis 




 1   .   3   .  .. PROOF OF THE VOLUME FORMULA. ..  17 

 
the last equality coming from di  ff  erentiating the identity




 open parenthesis   N,    X  sub  j   closing parenthesis   equiv   0 




in the  i  th direction. In particular, it follows from  open parenthesis   1   .   3   closing parenthesis  and the equality of cross

 
derivatives that




 open parenthesis   W  sub  x   X  sub  i   ,   X  sub  j   closing parenthesis   =   open parenthesis   X  sub  i   ,   W  sub  x   X  sub  j   closing parenthesis 




and hence, by linearity that




 open parenthesis   W  sub  x   u,    v   closing parenthesis   =   open parenthesis   u,    W  sub  x   v   closing parenthesis   forall   u,    v   in   TY  sub  x   . 




We have proved that the second fundamental form is symmetric, and hence the

 
Weingarten map is diagonizable with real eigenvalues.





Recall that the principal curvatures are, by definition, the eigenvalues of the

 
Weingarten map. We will let




 W   =   open parenthesis   W  sub  ij   closing parenthesis 




denote the matrix ofthe Weingarten map with respect to the basis  X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   . 

 
Explicitly,




Line 1  N  sub  i   =   sum   W  sub  ji   X  sub  j   .  Line 2  j 




If we write  N  sub  1   ,   .   .   .   ,   N  sub  n   minus   1   ,   X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1  as column vectors of length  n   ,  we can

 
write the preceding equation as the matrix equation




Equation:  open parenthesis   N  sub  1   ,   .   .   .   ,   N  sub  n   minus   1   closing parenthesis   =   open parenthesis   X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   closing parenthesis   W.  .. open parenthesis   1   .   4   closing parenthesis 




The matrix multiplication on the right is that of an  n   times   open parenthesis   n   minus   1   closing parenthesis  matrix with an

 
 open parenthesis   n   minus   1   closing parenthesis   times   open parenthesis   n   minus   1   closing parenthesis  matrix. To understand this abbreviated notation, let us write





it out in the case  n   =   3   ,  so that  X  sub  1   ,   X  sub  2   ,   N  sub  1   ,   N  sub  2  are vectors in  R  to the power of  3   : 




 X  sub  1   =  Row 1  X  sub  11  Row 2  X X  12  sub  13 .  ,   X  sub  2   =  Row 1  X  sub  21  Row 2  X X  22  sub  23 .  ,   N  sub  1   =  Row 1  N  sub  11  Row 2  N  sub  12 N  13 .  ,   N  sub  2   =  Row 1  N  sub  21  Row 2  N  sub  22 N  23 .  . 




Then  open parenthesis   1   .   4   closing parenthesis  is the matrix equation




 parenlefttp-parenleftbt   N   N   N  sub  13   12   11   N   N   N  sub  23   22   21   parenrightbt-parenrighttp   =   parenlefttp-parenleftbt   X   X  to the power of  X   11   13  to the power of  12   X   X  to the power of  X   21   23  to the power of  22   parenrightbt-parenrighttp   parenleftbigg   W   W  sub  21   11   W   W  sub  22   12   parenrightbigg   . 




Matrix multiplication shows that this gives




 N  sub  1   =   W  sub  11   X  sub  1   plus   W  sub  21   X  sub  2   ,   N  sub  2   =   W  sub  12   X  sub  1   plus   W  sub  22   X  sub  2   , 




and more generally that  open parenthesis   1   .   4   closing parenthesis  gives  N  sub  i   =   sum  sub  j   W  sub  ji   X  sub  j  in all dimensions.





Now consider the region  Y  sub  h   ,  the thickened hypersurface, introduced in the

 
preceding section except that we replace the full hypersurface  Y  by the portion

 
 X   open parenthesis   M   closing parenthesis   .  Thus the region in space that we are considering is




 open brace   X   open parenthesis   y   closing parenthesis   plus   lambda   N   open parenthesis   y   closing parenthesis   ,   y   in   M   ,   0   less   lambda   less or equal    h   closing brace   . 




 18  .. CHAPTER  1   .  .. THE PRINCIPAL CURVATURES.

 
It is the image of the region  M   times   open parenthesis   0   ,   h   closing square bracket   subset   R  to the power of  n   minus   1   times  R under the map




 open parenthesis   y,    lambda   closing parenthesis   mapsto-arrowright   X   open parenthesis   y   closing parenthesis   plus   lambda   N   open parenthesis   y   closing parenthesis   . 




We are assuming that this map is injective. .... By  open parenthesis   1   .   4   closing parenthesis   ,  it has Jacobian matrix




Line 1  open parenthesis   di   ff   erential   closing parenthesis  Line 2  J   =   open parenthesis   X  sub  1   plus   lambda   N  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   plus   lambda   N  sub  n   minus   1   ,   N   closing parenthesis   =  Line 3  open parenthesis   X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   ,   N   closing parenthesis   parenleftbigg   open parenthesis   I  sub  n   minus   1   plus   0   lambda   W   closing parenthesis   0 1  parenrightbigg   .   open parenthesis   1   .   5   closing parenthesis 




The right hand side of  open parenthesis   1   .   5   closing parenthesis  is now the product of two  n  by  n  matrices.

 
The change of variables formula in several variables says that




Equation:  V  sub  n   open parenthesis   h   closing parenthesis   =   integral  sub  M   integral  sub  0   h   bar   determinant   J   bar   dhdy  sub  1   times   times   times   dy  sub  n   minus   1   . .. open parenthesis   1   .   6   closing parenthesis 




Let us take the determinant of the right hand side of  open parenthesis   1   .   5   closing parenthesis   .  The determinant

 
of the matrix  open parenthesis   X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   ,   N   closing parenthesis  is just the  open parenthesis  oriented  closing parenthesis   n  dimensional volume of

 
the parallelepiped spanned by  X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   ,   N   .  .. Since  N  is of unit length and

 
is perpendicular to the  X  to the power of  prime  s, this is the same as the  open parenthesis  oriented  closing parenthesis   n   minus   1  dimensional

 
volume of the parallelepiped spanned by  X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   .  Thus,  quotedblleft  by definition  quotedblright   , 




Equation:  bar   determinant   open parenthesis   X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   ,   N   closing parenthesis   bar   dy  sub  1   times   times   times   dy  sub  n   minus   1   =   d  to the power of  n   minus   1   A.  .. open parenthesis   1   .   7   closing parenthesis 




 open parenthesis  We will come back shortly to discuss why this is the right definition.  closing parenthesis  .. The

 
second factor on the right hand side of  open parenthesis   1   .   5   closing parenthesis  contributes




 determinant   open parenthesis   1   plus   lambda   W   closing parenthesis   =   open parenthesis   1   plus   lambda   k  sub  1   closing parenthesis   times   times   times   open parenthesis   1   plus   lambda   k  sub  n   minus   1   closing parenthesis   . 




For sufficiently small  lambda   ,  this expression is positive, so we need not worry about

 
the absolute value sign if  h  small enough. Integrating with respect to  lambda  from  0 




 to   h   gives   open parenthesis   1   .   2   closing parenthesis   . 




We proved  open parenthesis   1   .   2   closing parenthesis  if we define  d  to the power of  n   minus   1   A  to be given by  open parenthesis   1   .   7   closing parenthesis   .  But then it follows




Line 1  from   open parenthesis   1   .   2   closing parenthesis   that  Line 2  d  divided by  dh   V  sub  n   open parenthesis   Y  sub  h   closing parenthesis  sub  bar   h   =   0   =   integral  sub  Y   d  to the power of  n   minus   1   A.    open parenthesis   1   .   8   closing parenthesis 




A moment  quoteright  s thought shows that the left hand side of  open parenthesis   1   .   8   closing parenthesis  is exactly what we

 
want to mean by  quotedblleft  area  quotedblright   :  it is the  quotedblleft  volume ofan infinitesimally thickened region  quotedblright   . 

 
This justifies taking  open parenthesis   1   .   7   closing parenthesis  as a definition. .. Furthermore, although the definition

 
 open parenthesis   1   .   7   closing parenthesis  is only valid in a coordinate neighborhood, and seems to depend on the

 
choice of local coordinates, equation  open parenthesis   1   .   8   closing parenthesis  shows that it is independent of the

 
local description by coordinates, and hence is a well defined object on  Y   .  .. The

 
functions  H  sub  j  have been defined independent of any choice of local coordinates.





Hence  open parenthesis   1   .   2   closing parenthesis  works globally  :  .. To compute the right hand side of  open parenthesis   1   .   2   closing parenthesis  we may

 
have to break  Y  up into patches, and do the integration in each patch, summing

 
the pieces. But we know in advance that the final answer is independent of how

 
we break  Y  up or which local coordinates we use.





 1   .   4   .  .. GAUSS  quoteright  S THEOREMA EGREGIUM. ..  19 

 
 1   .   4  .. Gauss  quoteright  s .. theorema egregium.





Suppose we consider the two sided region about the surface, that is




 V  sub  n   open parenthesis   Y   plus   h   closing parenthesis   plus   V  sub  n   open parenthesis   Y   minus   h   closing parenthesis 




corresponding to the two di  ff  erent choices of normals. When we replace  nu   open parenthesis   x   closing parenthesis  by





 minus   nu   open parenthesis   x   closing parenthesis  at each point, the Gauss map  nu  is replaced by  minus   nu   ,  and hence the Wein  hyphen 

 
garten maps  W  sub  x  are also replaced by their negatives. The principal curvatures

 
change sign. .. Hence, in the above sum the coefficients of the even powers of  h 

 
cancel, since they are given in terms ofproducts ofthe principal curvatures with

 
an odd number of factors. .. For  n   =   3  we are left with a sum of two terms, the





coefficient of  h  which is the area, and the coefficient of  h  to the power of  3  which is the integral

 
of the Gaussian curvature. .. It was the remarkable discovery of Gauss that this

 
curvature depends only on the intrinsic geometry of the surface, and not on

 
how the surface is embedded into three space. Thus, for both the cylinder and

 
the plane the cubic terms vanish, because  open parenthesis  locally  closing parenthesis  the cylinder is isometric to

 
the plane. .. We can wrap the plane around the cylinder without stretching or

 
tearing.





It was this fundamental observation ofGauss that led Riemann to investigate

 
the intrinsic metric geometry of higher dimensional space, eventually leading

 
to Einstein  quoteright  s general relativity which derives the gravitational force from the

 
curvature of space time. .. A first objective will be to understand this major

 
theorem of Gauss.





An important generalization of Gauss  quoteright  s result was proved by Hermann Weyl

 
in  1939   .  .. He showed  :  .. if  Y  is any  k  dimensional submanifold of  n  dimensional

 
space  open parenthesis  so for  k   =   1   ,   n   =   3   Y  is a curve in three space  closing parenthesis   ,  let  Y   open parenthesis   h   closing parenthesis  denote the

 
 quotedblleft  tube  quotedblright  around  Y  of radius  h   ,  the set of all points at distance  h  from  Y   .  Then,

 
for small  h   ,   V  sub  n   open parenthesis   Y   open parenthesis   h   closing parenthesis   closing parenthesis  is a polynomial in  h  whose coefficients are integrals over

 
 Y  of intrinsic expressions, depending only on the notion of distance within  Y   . 





Let us multiply both sides of  open parenthesis   1   .   4   closing parenthesis  on the left by the matrix  open parenthesis   X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   closing parenthesis  to the power of  T 

 
to obtain




 L   =   QW 




where  L  sub  ij   =   open parenthesis   X  sub  i   ,   N  sub  j   closing parenthesis  as before, and




 Q   =   open parenthesis   Q  sub  ij   closing parenthesis   :   =   open parenthesis   X  sub  i   ,   X  sub  j   closing parenthesis 




is called the matrix of the first fundamental form relative to our choice of

 
local coordinates. All three matrices in this equality are of size  open parenthesis   n   minus   1   closing parenthesis   times   open parenthesis   n   minus   1   closing parenthesis   . 





If we take the determinant of the equation  L   =   QW  we obtain




Equation:  determinant   W   =   determinant   L  divided by  determinant   Q   , .. open parenthesis   1   .   9   closing parenthesis 




an expression for the determinant of the Weingarten map  open parenthesis  a geometrical prop  hyphen 

 
erty of the embedded surface  closing parenthesis  as the quotient of two local expressions. For the

 
case  n   minus   1   =   2   ,  we thus obtain a local expression for the Gaussian curvature,




 K   =   determinant   W   . 




 20  .... CHAPTER  1   .  .... THE PRINCIPAL CURVATURES.





The first fundamental form encodes the intrinsic geometry of the hypersur  hyphen 

 
face in terms oflocal coordinates  :  it gives the Euclidean geometry ofthe tangent

 
space in terms of the basis  X  sub  1   ,   .   .   .   ,   X  sub  n   minus   1   .  .. If we describe a curve  t   arrowright-mapsto   gamma   open parenthesis   t   closing parenthesis  on





the surface in terms of the coordinates  y  to the power of  1   ,   .   .   .   ,   y  to the power of  n   minus   1  .... by giving the functions





 t   arrowright-mapsto   y  to the power of  j   open parenthesis   t   closing parenthesis   ,   j   =   1   ,   .   .   .   ,   n   minus   1  then the chain rule says that




 gamma  to the power of  prime   open parenthesis   t   closing parenthesis   =   sum  from  j   =   1  to  n   minus   1   X  sub  j   open parenthesis   y   open parenthesis   t   closing parenthesis   closing parenthesis   dy  to the power of  j  divided by  dt   open parenthesis   t   closing parenthesis 




where




 y   open parenthesis   t   closing parenthesis   =   open parenthesis   y  to the power of  1   open parenthesis   t   closing parenthesis   ,   .   .   .   ,   y  to the power of  n   minus   1   open parenthesis   t   closing parenthesis   closing parenthesis   . 




Therefore the  open parenthesis  Euclidean  closing parenthesis  square length of the tangent vector  gamma  to the power of  prime   open parenthesis   t   closing parenthesis  is




 bar   gamma  to the power of  prime   open parenthesis   t   closing parenthesis   bar  to the power of  2   =   sum  from  i, j   =   1  to  n   minus   1   Q  sub  ij   open parenthesis   y   open parenthesis   t   closing parenthesis   closing parenthesis   dy  to the power of  i  divided by  dt   open parenthesis   t   closing parenthesis   dy  to the power of  j  divided by  dt   open parenthesis   t   closing parenthesis   . 




Thus the length of the curve  gamma  given by




 integral   bar   gamma  to the power of  prime   open parenthesis   t   closing parenthesis   bar   dt 




can be computed in terms of  y   open parenthesis   t   closing parenthesis  as




 integral  square root of  sum  from  i, j   =   1  to  n   minus   1   Q  sub  ij   open parenthesis   y   open parenthesis   t   closing parenthesis   closing parenthesis   dy  to the power of  i  divided by  dt   open parenthesis   t   closing parenthesis   dy  to the power of  j  divided by  dt   open parenthesis   t   closing parenthesis   dt 




 open parenthesis  so long as the curve lies within the coordinate system  closing parenthesis   . 





So two hypersurfaces have the same local intrinsic geometry if they have the

 
same  Q  in any local coordinate system.





In order to conform with a  open parenthesis  somewhat variable  closing parenthesis  classical literature, we shall

 
make some slight changes in our notation for the case of surfaces in three di  hyphen 

 
mensional space. .. We will denote our local coordinates by  u,    v  instead of  y  sub  1   ,   y  sub  2 

 
and so  X  sub  u  will replace  X  sub  1  and  X  sub  v  will replace  X  sub  2   ,  and we will denote the scalar

 
product of two vectors in three dimensional space by a  times  instead of  open parenthesis   ,   closing parenthesis   .  .. We

 
write




Equation:  Q   =  Row 1  E   F  Row 2  F   G ... open parenthesis   1   .   10   closing parenthesis 




where




Equation:  E   :   =   X  sub  u   times   X  sub  u .. open parenthesis   1   .   11   closing parenthesis 
Equation:  F   :   =   X  sub  u   times   X  sub  v .. open parenthesis   1   .   12   closing parenthesis 
Equation:  G   :   =   X  sub  v   times   X  sub  v .. open parenthesis   1   .   13   closing parenthesis 




so




Equation:  determinant   Q   =   EG   minus   F  to the power of  2   . .. open parenthesis   1   .   14   closing parenthesis 
