B.
Local Coordinates,
Curvature,
and Area
I CHP a surface
I : V → I a coordinate chart,
V c IN'
for any I E 1mF we saw in section HI that
Tp -2 = Im DIE where I EV s # Itg )ip
It we use coordinates luv ) on Vc IR'
then Im Dtp is spanned by
Fiel and 3747
we denote these by
Tip ) and Ttp)
recall we THE
a::÷÷. : '
(9) by Iv
The 1st fundamental form of I C IR'
on Tp -2 is
Iip ) = Tp -2 x Tp -2 → RU
⇐, Tra ) 1-7 F. WT
Iers in IR
'
Tdot product in 473
so I Ip ) is an inner product on Tp -2
if we set a Cpt = I Cpt . I Ip )
b CF) = it Lp ) . Tip )
c Cp) = ftp.vlp )
then I Cpt can be represented by the matrix
a Cp) b C p )
( bird cops)
to see this note any vectors I, I C- Tp -2 can be written
I = x,
a- txzv = I# in the basis it
,I
I -
- y ,I tyzv = [ Y
,:]now I CptII Ty ) = I . I
= ( x ,I exit ) . ( y ,
I t yet )
= x, y ,
a t X, Yztxzy ,) b t X z Yz C
-
- Exist lab:) I
so in the basis a,
I
Iim -
- I 997, %:3)and this measures lengths of vectors and anglesbetween vectors in Tp -2
now for g- in V define
gig) : 1122×1122 → IR
by gig) I wi . %)= I HTT) ) ( Dfg Cut )
, Dfg Cod )
= LDF-quiDo ( Dig CE ) ]
so we can represent g in the basis Fu,Zo by
a. tan: . it¥:DIdea is g represents I in local coordinates
I : V → E
we callg a Riemannian metric ( but for Ic IR
'it is essentially
equivalent to the 1st fundamental form,
later )
many computations on I can be done in local coordinates
using g Icu)
example . I-€7i. I
1-1 F -- to I
a b
length (pi ) = fall f' Hill It
=Sab IlfEt ) d t
=SabIttDDzIDDz⇐I 'HD dt
= Jab
g.CI#)lEHI.I'HTdt--baHI'ttNggdtmeans length of
veto r using g
Recall from calc HI :
if a surface I in IRS is parameterized byI : V → IR
3
Vc 1122
( u, v )
and R -
- I ( v ) a -2
then the area of R is
Area C R ) = fullTill dado
where it = Futit = Iot
note : HaxTH = KUTIHEH sin 0 ¥,%= kill HEH 1-6507
-
- thinkinf-fEi=
HuTTuvTP_µ=ac-bT
= deft = detg
so Area IR ) = Svidetgdudr
example : 5 c 1123 be the unit sphere
I : y → 5 : @ioTi-sfu.v,Fut)
( I
{ Cair ) E IR ? u 't v'
c I }
so a- Eat -
- toe If ] to ] -1¥)
similarly of =/
"
a -- ii. a = It 7¥ -
- IET
b = T.ci = Url- U2 - ✓
2
C = T . I =
I - U2 - V2
and
gun , = IECIiitlet U -
- upper hemisphere of 53 so
area ( U ) = £,
idetg du do
= f,
,(FyzH- wth - rt - u 're)hedv= S
, ,
Cl - on - v)"
Z
dado
= J,
deed v
change →
to polar = J, ,
rdr do
word - s
= Sis ! door
= ZIT f - if ) to = 2T
Recall to compute Gauss and mean curvature we can take
the normal vector it to the surface and consider
the shape operator
Sp CE ) -- - to 457
then the Gauss curvature is Kip ) = def Sp
and the mean curvature is Hip ) = I tr Sp} " G " 5
now if I :X → E gives local coordinates on -2 then
Tp -2 is spanned by it = (Dfg ) ⇐ ) where Feit =p
ri -- DIE ) Br )
and
I =u
the XIII
in the basis Tiv for Tp E we can write Sp as
Sp-
- Laa!,
9
a;) just as we did above
before lemma 3
we need to find aij
Set A = Split . I
B = Sp tu ) . I
C-= Sp IT ) . I
Remark : The 2nd fundamental form of E atp is
I Cpt :Tp Ex Tp -2 → IR
④ , WI) t Sp I Tv,
) . I
so in the basis air we can represent Itp ) as
( AB Be ) just like we did for the
1st fundamental form above
Spi and II Ipt have the same information,
so I only mention
this since some books prefer I and some S
note : A = Sp tu ) . I = I ai ,T t
az ,I ) . I = a
, ,a tar ,
b
B = Spca ) . I = ( a, ,
I taut ) . F = a. ,b t q ,
c
Sp I T ) .I = ( aizu t ant ) . I = a. za t an b
C = Sp ( I ) . I = ( aizu tazz I) . I = a, z
b t an C
this is equivalent to saying
*:3 .
. la:" a:X ::3or Ca:: 'Il ::S
"
= .tl:71 ⇐is ]so we have proven the followinglemma 6 :
with the above notation
Ktp ) = def Sp = deftlydef Iip )
= ACa c - b -
and
Hip ) = Etr Sp= Iz Ac-2bBt#
ac - 62
Remark : In the proof of lemma 2 we saw that
(E) . a- =- T . Faa
Wolf -
-
- t.FI ,
City ) . -v= - t . too
where Fascist = LD ( DIE I Eu )),
# ) ] for g- st.
I toil=p
this can be very helpful when computingexample a > r so fixed constants
I I u.
v ) = ( ( atr cos u ) cosy ( a tr cos u ) sin v,
r sin u )
ZIT
i.torus ( donut )
vi. Ia = ( - rsinucosv,
- r sinus inv,
roosa )
of = If = flatroosa ) sin v,
Latrcosu ) cosy o )
SO : a = I. I = r2
b = vi. T = O
c = catrosa ,a
9 -
- ⑨be )
and
N- = Y¥#r,=
. . .
= - ( cos u cost, cos a Shiv
,Smu )
w
exercise
and
Faa = I - rcosucosv,
- rcosusinv,
- r since )
Ffa = ( r sinus inv,
- rsinucosv ,O )
Izz = ( - I at rosa ) loser,
- ( at rios a) Sisu,
o )
SO :
A = Stu ) . it = F.
Faa= r
B -
- Stu ) . I = I .
Taz = O
C -
- Shh . T = F. try = ( at rcosu ) cos u
Has :
kcu.ir ) det I=
AC - B'
=rcatrcosu ) cos u
- -def I ac - b'
r- ( atr cosa )
2
= Losar( at r cos a )
v Kip )
+ - + →
I 'Iz u
C. Some Implications of Curvature
note Sp is particularly scaiple if K,
= KE C
i.e. Split ) = CT
we call such a point is an um bi tic pointThat .
If every point of E c 1123 is umbilici then
I is a part of a plane if K = O or
I is a part of a sphere of radius it K > o
( need I path connected for this )
note the Gauss curvature at an umbilici point is 20
Proof : let I :all,uj→E c IR
'be a word
.chart
set a = Diner,I Iu )
T =D Far,( ⇒ } span Team,
-2
since all points are umbilici we have
$ might →K I = Sp tu ) = - NIU
depend -, KJ = Spelt ) - - Tv
on IIn the proof of lemma 2.3 we saw
( to) ,
=
,)
,
we actually did this for Ibut same computationworks for T too
SO
Half -
- Nth
Ky I t K Tiz = KIT t K Fa
but recall to - (DIfIuL)z= DID In.
.nl#Dfiu..fZr)j-DlDttu.nlErDfcu.nlFu)
lemma 2.3
computation =
FasoKy I = Kiev
and since I and of are linearly independent we see
Kg = Kui = O
so K is constant on coordinate charts
exercise . Show K constant on all of -2
Hint : pig E I take path I from p IoT
cover link ) by charts and consider
overlap of charts
Case I : K -
- o,
then E part of a plane
Proofs: K -
- O ⇒ Safir ) = O fir and p
so Ff (f) =-
Sp CF ) = o tf E and p
Intuitively,
it not changing so I must be in a-
plane perpendicular to I
Rigorously: recall the equation for a plane perp .
tot is
a. o.o
,
and the plane perp to I but through I is
the set of all I satisfying,
of
f. co - pt = o
for a fixed FEE let D= Tcp )
now for any g- EE,
let I :{ o ,c ] → I be an
arc with I cos =p and ICD= q
Set f- C t ) = ( ICH - p ) . N' I ICH )
f'
It ) = It tho NTI KD t I ICH - p ) . D Faye,
LI ' CED- -
= O since I'M c- Tache - S, #
( I 'It )
= - ( Ict ) - p ) o S, #
l I 'It ) )
= O
so fit ) is constant !
f- Co) = ( F - p ) . Dcp ) = o
i. fu ) = O
I I
(g- - p ) . N' Cg) but It NTI KD = DNF#CI '
HDit
I = - S, #
( Ita )= O
IF-F) ' Iso NTI ) = Bcp) =D
So for any point g- EE is in the plane
(Ep
) . it to
Case 2 : K to
Proof: let I : V → E be a word chart
Consider : Flav) = I luv ) txt NTI Cair )-
.
note ¥ Flair ) = Father ) txt I Diff.
. . ,) I Fat curl )
=
Futian- ¥ Spca .nl Father )
- -
= Zu f - ¥ Eat = O
similarly ZrF=o
so F is constant, say F- Car ) = I
that is,
we have I luv ) t INT Fears) = E
so It Flair ) - Elle It - ¥NTfTuvDH = ¥
that is any point in the image of I is on
the sphere of radius I about I
the Gauss curvature is K = KK
so the radius is
exercise See other word charts for -2 are
on the same sphereL#
That 8 .
If I E IRS is a compact surface,
then there
is some point § E E such that Kip ) > o
7 roof : consider f : I → IR ip-llplf-p.ptf- is continuous
a result from calculus implies that any continuous
function on a compact set has a maximum Cand mint
so I fo E -2 such that f- I p ) E f Cfo ) V p E E
..
siriasis:.
Ideas. at Do,
E curves more ( in all directions ) than
the sphere of radius It pill about the org in
so all principal curvatures are larger than
those of the sphere ⇐1¥ )
so we expect K Cpo )z¥p > 0
Tomakethisrigorous.com pate the curvature of
-2 at PJ in direction I E Up -2
recall,
there is a curve
I :L - E. e ] → E
such that I co ) =
ToI 'co ) = I
HI ' kN =L htt
from lemma I we know← normal to E
K pot) = I "
lol . N' Cfo )
now lets fail it
for any tangent vector E e Tpo -2 let
§ be a path in I set.
§ Lot Fo
§'co ) = I
not : fo B- It ) = B-Lt ) . Jlt )
So dat top It ) I⇐ o
-
- F'
to ) . B- Colt I Co) - Bio)
= 2p-o.to
but top has a maximum at O ( since f- on
-2 does )
so 2 Do . I = ¥ top # o
= O
re.
E is perpendicular to Fo VT E TEEthus D= ftp.T,
is ( or- this
,but sign wont
matter )
now consider fo I = I . I
since
⇐o is a maximum of fo I we see
dat fo I # 0=0
and
d¥ fo Itt= o
£0
OZ htt Ifee.I .IT/o=dzl2I' E) to
= 2K".It I !I ' ) to
= 2 ( I"
to ) . Fo t I )
so PJ . I' '
to ) E - I
and Kpocci ) = I' '
co ) . I = I " Colo PI" Doll
± -
upto < 0
for any I E Up.
-2
so both principal curvatures are a- t
- ltpoll
>I
> o:.
K Ipo ) = Kika- ltpoll
'
#