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Differential Geometry > epiphany > Flashcards

epiphany Flashcards

1
Q

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

A

unit normal vector perpendicular to tangent space of surface at point

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2
Q

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

A

smooth map between surface

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3
Q

Derivative at point p of a smooth map between surfaces is

A

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)

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4
Q

d_pN is a linear map from T_pS into

A

itself (endomorphism)

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5
Q

Weingarten map is

A

-d_pN

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6
Q

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

A

nbilinnear form <.,.>_p

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7
Q

-d_pN can be represented by a

A

symmetric matrix with respect to an orthonormal basis of T_pS

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8
Q

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

A

real and eigenvectors of diff eigenvalues of symmetric matrix are perpendicular

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9
Q

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

A

basis of T_pS

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10
Q

matrix is symmetric then A^T =

A

A

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11
Q

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

A

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

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12
Q

Eignevalues, trace and determinant do not depend on

A

choice of basis

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13
Q

A and A’ are linked by

A

A’ = B^(-1)AB

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14
Q

Trace of a matrix is the

A

sum of its diagonal entries and sum of its eigenvalues

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15
Q

determinant can also be

A

product of its eigenvalues

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16
Q

Principal curvatures of S at p are

A

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

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17
Q

The principal direction of S at p=

A

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

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18
Q

Gauss curvature is the product of the

A

principal curvatures

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19
Q

Mean curvature is the

A

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

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20
Q

We have positive gauss curvature if both principal curvatures

A

have the same sign

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21
Q

det of product =

A

product of determinants

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22
Q

det of inverse of matrix A =

A

1/det A

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23
Q

H(p) =

A

1/2 * sum of diagonal matrices

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24
Q

A local parametrisation with F= 0 is called

A

orthogonal

25
Q

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

A

principal

26
Q

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

A

> 0

27
Q

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

A

< 0

28
Q

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

A

= 0

29
Q

the subset K(p)>0 are called

A

elliptic regions of S

30
Q

the subset K(p)<0 are called

A

hyperbolic region of S

31
Q

the subset K(p) = 0 are called

A

flat region of S

32
Q

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

A

k1 = k2 = 0

33
Q

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

A

k1 = k2

34
Q

local parametrisation with F= 0 is

A

orthogonal

35
Q

local parametrisation with F= 0 and M= 0 is

A

principal

36
Q

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

A

L/E AND N/G

37
Q

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

A

minimal surface

38
Q

a metric space is complete if every cauchy sequence is

A

convergent in surface

39
Q

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

A

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

40
Q

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 =

A

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

41
Q

every compacts surface hs at least one …. point

A

elliptic

42
Q

there are no compact surfaces which are

A

minimal

43
Q

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

A

sphere

44
Q

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

A

plane

45
Q

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

A

umbilic points

46
Q

Willmore function does not change under rescaling because

A

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

47
Q

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

A

round sphere

48
Q

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

A

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

49
Q

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

A

1/r^2 > 0

50
Q

Euclidean planes have constant gauss curvature k =

A

0

51
Q

hyperbolic planbe has constant gauss curvature k =

A

-1

52
Q

gauss curvature is preserved under

A

local isomteries

53
Q

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

A

acceleartion vector alpha’’

54
Q

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

A

normal curvature (Meusnier)

55
Q

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

A

minimum and maximum

56
Q

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

A

eigenvector of the weingarten map for all t

57
Q

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

A

eigenvector of the weingarten map

58
Q

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

A

lies of curvature

59
Q

a curve on a regular surface is asymptotic curve if its

A

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

Differential Geometry (2 decks)

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