epiphany

Question 1

Gauss map is a smooth map associating to every point p on a regular surface a

Answer 1

unit normal vector perpendicular to tangent space of surface at point

Question 2

since normal vectors lie in the unit sphere the Guass map is a

Answer 2

smooth map between surface

Question 3

Derivative at point p of a smooth map between surfaces is

Answer 3

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)

Question 4

d_pN is a linear map from T_pS into

Answer 4

itself (endomorphism)

Question 5

Weingarten map is

Answer 5

-d_pN

Question 6

Weingarten map, a linear map of vector space T_pS is symmetric with respect to the

Answer 6

nbilinnear form <.,.>_p

Question 7

-d_pN can be represented by a

Answer 7

symmetric matrix with respect to an orthonormal basis of T_pS

Question 8

Eignevalues and eigenvectors of symmetric matrices (b=c) are all

Answer 8

real and eigenvectors of diff eigenvalues of symmetric matrix are perpendicular

Question 9

Weingarten map can be represented by a matric once we have chosen a

Answer 9

basis of T_pS

Question 10

matrix is symmetric then A^T =

Answer 10

A

Question 11

since - d_pN is symmetric, A_ij = <-d_pN(w_j),w_i>_p =

Answer 11

= <-d_pN(w_i),w_j>_p = A_ji

Question 12

Eignevalues, trace and determinant do not depend on

Answer 12

choice of basis

Question 13

A and A’ are linked by

Answer 13

A’ = B^(-1)AB

Question 14

Trace of a matrix is the

Answer 14

sum of its diagonal entries and sum of its eigenvalues

Question 15

determinant can also be

Answer 15

product of its eigenvalues

Question 16

Principal curvatures of S at p are

Answer 16

eigenvalues k1(p),k2(p) of -d_pN

Question 17

The principal direction of S at p=

Answer 17

eigenvectors X1(p),X2(p) of -d_pN

Question 18

Gauss curvature is the product of the

Answer 18

principal curvatures

Question 19

Mean curvature is the

Answer 19

average of the principal curvatures (SUM AND DIVIDE BY 2)

Question 20

We have positive gauss curvature if both principal curvatures

Answer 20

have the same sign

Question 21

det of product =

Answer 21

product of determinants

Question 22

det of inverse of matrix A =

Answer 22

1/det A

Question 23

H(p) =

Answer 23

1/2 * sum of diagonal matrices

Question 24

A local parametrisation with F= 0 is called

Answer 24

orthogonal

Question 25

A local parametrisation with F=0 and M=0 is called

Answer 25

principal

Question 26

S is a regular surface with a Gauss map. A point is elliptic if K(p)

Answer 26

>0

Question 27

S is a regular surface with a Gauss map. A point is hyperbolic if K(p)

Answer 27

< 0

Question 28

S is a regular surface with a Gauss map. A point is flat if K(p)

Answer 28

= 0

Question 29

the subset K(p)>0 are called

Answer 29

elliptic regions of S

Question 30

the subset K(p)<0 are called

Answer 30

hyperbolic region of S

Question 31

the subset K(p) = 0 are called

Answer 31

flat region of S

Question 32

A point is called a planar point of the surface S if

Answer 32

k1 = k2 = 0

Question 33

A point p is called an umbilic point of the surface if

Answer 33

k1 = k2

Question 34

local parametrisation with F= 0 is

Answer 34

orthogonal

Question 35

local parametrisation with F= 0 and M= 0 is

Answer 35

principal

Question 36

if regular surface is principal then X_u and x_v are principal directions with k1 and k2 =

Answer 36

L/E AND N/G

Question 37

a regular surface with everywhere vanishing mean curvature (H = 0) is called a

Answer 37

minimal surface

Question 38

a metric space is complete if every cauchy sequence is

Answer 38

convergent in surface

Question 39

cauchy sequence means for every epsilon > 0 there exists an N in N such that

Answer 39

d(pn,pm) < epsilon for all n,m>=N

Question 40

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 =

Answer 40

sup ds(p1,p2) <= pi/sqrt(K_0)

Question 41

every compacts surface hs at least one .... point

Answer 41

elliptic

Question 42

there are no compact surfaces which are

Answer 42

minimal

Question 43

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

Answer 43

sphere

Question 44

f all points of regular connected surface are planar then S is part of a

Answer 44

plane

Question 45

if surface is a convex regular surface diffeomorphic to a sphere then it has at least 2

Answer 45

umbilic points

Question 46

Willmore function does not change under rescaling because

Answer 46

surface is bigger but H is smaller so balances out curvautre integrating smaller function over bigger surface

Question 47

W(s) >= 4pi iff S is a

Answer 47

round sphere

Question 48

theorema egregium: the gauss curvature at a point, p,of a regular surface depends only on .....

Answer 48

the coefficients of E,F,G and their derivatives of a local parametrisation

Question 49

Round spheres of radius r>0 have constant gauss curvature k =

Answer 49

1/r^2 > 0

Question 50

Euclidean planes have constant gauss curvature k =

Answer 50

0

Question 51

hyperbolic planbe has constant gauss curvature k =

Answer 51

-1

Question 52

gauss curvature is preserved under

Answer 52

local isomteries

Question 53

smooth unit speed space curve the unit tangent vector (t = alpha') is perpendicular to the

Answer 53

acceleartion vector alpha''

Question 54

all reuglar curves in a surface through a point with the same tangent vector have the same

Answer 54

normal curvature (Meusnier)

Question 55

k1 and k2 are the ...... of possible normal curvatures obtained via regular curves

Answer 55

minimum and maximum

Question 56

a regular curve is a line of curvature if alpha' is an

Answer 56

eigenvector of the weingarten map for all t

Question 57

curve is called line of curvature if alpha'(t) is an

Answer 57

eigenvector of the weingarten map

Question 58

for a prinicpal parametrisation the coordinate curves, (u-> x(u,v 0) and v -> x(uo, v) are

Answer 58

lies of curvature

Question 59

a curve on a regular surface is asymptotic curve if its

Answer 59

normal vanishes identically that is normal curvature (k_n) = 0