epiphany Flashcards
Gauss map is a smooth map associating to every point p on a regular surface a
unit normal vector perpendicular to tangent space of surface at point
since normal vectors lie in the unit sphere the Guass map is a
smooth map between surface
Derivative at point p of a smooth map between surfaces is
a linear map from tangent space of surface at p to tangent space of unit sphere at point N(p) (N(p) is gauss map at point p)
d_pN is a linear map from T_pS into
itself (endomorphism)
Weingarten map is
-d_pN
Weingarten map, a linear map of vector space T_pS is symmetric with respect to the
nbilinnear form <.,.>_p
-d_pN can be represented by a
symmetric matrix with respect to an orthonormal basis of T_pS
Eignevalues and eigenvectors of symmetric matrices (b=c) are all
real and eigenvectors of diff eigenvalues of symmetric matrix are perpendicular
Weingarten map can be represented by a matric once we have chosen a
basis of T_pS
matrix is symmetric then A^T =
A
since - d_pN is symmetric, A_ij = <-d_pN(w_j),w_i>_p =
= <-d_pN(w_i),w_j>_p = A_ji
Eignevalues, trace and determinant do not depend on
choice of basis
A and A’ are linked by
A’ = B^(-1)AB
Trace of a matrix is the
sum of its diagonal entries and sum of its eigenvalues
determinant can also be
product of its eigenvalues
Principal curvatures of S at p are
eigenvalues k1(p),k2(p) of -d_pN
The principal direction of S at p=
eigenvectors X1(p),X2(p) of -d_pN
Gauss curvature is the product of the
principal curvatures
Mean curvature is the
average of the principal curvatures (SUM AND DIVIDE BY 2)
We have positive gauss curvature if both principal curvatures
have the same sign
det of product =
product of determinants
det of inverse of matrix A =
1/det A
H(p) =
1/2 * sum of diagonal matrices
A local parametrisation with F= 0 is called
orthogonal
A local parametrisation with F=0 and M=0 is called
principal
S is a regular surface with a Gauss map. A point is elliptic if K(p)
> 0
S is a regular surface with a Gauss map. A point is hyperbolic if K(p)
< 0
S is a regular surface with a Gauss map. A point is flat if K(p)
= 0
the subset K(p)>0 are called
elliptic regions of S
the subset K(p)<0 are called
hyperbolic region of S
the subset K(p) = 0 are called
flat region of S
A point is called a planar point of the surface S if
k1 = k2 = 0
A point p is called an umbilic point of the surface if
k1 = k2
local parametrisation with F= 0 is
orthogonal
local parametrisation with F= 0 and M= 0 is
principal
if regular surface is principal then X_u and x_v are principal directions with k1 and k2 =
L/E AND N/G
a regular surface with everywhere vanishing mean curvature (H = 0) is called a
minimal surface
a metric space is complete if every cauchy sequence is
convergent in surface
cauchy sequence means for every epsilon > 0 there exists an N in N such that
d(pn,pm) < epsilon for all n,m>=N
Bonnet-Myers theorem: Surface is connectd and regular which is a complete metric space. there is K(p)>= K_0 so surface has bounded diamater =
sup ds(p1,p2) <= pi/sqrt(K_0)
every compacts surface hs at least one …. point
elliptic
there are no compact surfaces which are
minimal
If all points of a connected regular set are umbilic and K(p) is not 0 in at least one point then surface is part of a
sphere
f all points of regular connected surface are planar then S is part of a
plane
if surface is a convex regular surface diffeomorphic to a sphere then it has at least 2
umbilic points
Willmore function does not change under rescaling because
surface is bigger but H is smaller so balances out curvautre integrating smaller function over bigger surface
W(s) >= 4pi iff S is a
round sphere
theorema egregium: the gauss curvature at a point, p,of a regular surface depends only on …..
the coefficients of E,F,G and their derivatives of a local parametrisation
Round spheres of radius r>0 have constant gauss curvature k =
1/r^2 > 0
Euclidean planes have constant gauss curvature k =
0
hyperbolic planbe has constant gauss curvature k =
-1
gauss curvature is preserved under
local isomteries
smooth unit speed space curve the unit tangent vector (t = alpha’) is perpendicular to the
acceleartion vector alpha’’
all reuglar curves in a surface through a point with the same tangent vector have the same
normal curvature (Meusnier)
k1 and k2 are the …… of possible normal curvatures obtained via regular curves
minimum and maximum
a regular curve is a line of curvature if alpha’ is an
eigenvector of the weingarten map for all t
curve is called line of curvature if alpha’(t) is an
eigenvector of the weingarten map
for a prinicpal parametrisation the coordinate curves, (u-> x(u,v 0) and v -> x(uo, v) are
lies of curvature
a curve on a regular surface is asymptotic curve if its
normal vanishes identically that is normal curvature (k_n) = 0
