Page 304 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY
P. 304
Unit 24: Two Fundamental Form
Proof. Let V . The unit normal N( ) of X at is perpendicular to T X. By Proposition 1, we Notes
have T ( ) X T X. Thus, N( ) is perpendicular to T ( ) X, as is ( ) . It follows that the two
N
vectors are co-linear, and hence N( ) N ( ) . But since is orientation preserving, the two
pairs (X , X ) and X ,X 2 have the same orientation in the plane T X. Since also, the two triples
1
1
2
X1 ( ) ,X2 ( ) ,N ( ) and X ( ),X ( ),N( ) 2 have the same orientation in , it follows
3
1
N
that ( ) N( ).
24.2 The First Fundamental Form
Definition 6. A symmetric bilinear form on a vector space V is function B : V V satisfying:
1. B(aX + bY,Z) = aB(X,Z) + bB(Y,Z), for all X, Y V and a, b R.
2. B(X, Y) = B(Y, X), for all X, Y V.
The symmetric bilinear form B is positive definite if B(X, X) 0, with equality if and only if
X = 0.
With any symmetric bilinear form B on a vector space, there is associated a quadratic form Q(X)
= B(X, X). Let V and W be vector spaces and let T : V W be a linear map. If B is a symmetric
bilinear form on W, we can define a symmetric bilinear form T* Q on V by T* Q(X, Y) = Q(TX,
TY). We call T* Q the pull-back of Q by T. The map T is then an isometry between the inner-
product spaces (V, T* Q) and (W,Q).
Example: Let V = and define B(X, Y) = X Y , then B is a positive definite symmetric
3
bilinear form. The associated quadratic form is Q(X) = |X| .
2
Example: Let A be a symmetric 2 × 2 matrix, and let B(X, Y) = AX Y, then B is a
symmetric bilinear form which is positive definite if and only if the eigenvalues of A are both
positive.
3
Definition 7. Let X : U be a parametric surface. The first fundamental form is the symmetric
bilinear form g defined on each tangent space T X by:
u
g(Y, Z) = Y Z, Y, Z T X.
u
Thus, g is simply the restriction of the Euclidean inner product in above Example to each tangent
space of X. We say that g is induced by the Euclidean inner product.
Let g = g(X , X), and let Y = y X and Z = z X be two vectors in T X, then
i
i
u
i
i
ij
j
i
g(Y,Z) = g y z. j ...(3)
i
ij
Thus, the so-called coordinate representation of g is at each point u U an instance of above
0
example. In fact, if A = (g ), and B(, ) = A for as in the above example, then B is the
ij
pull-back by dX : 2 T X of the restriction of the Euclidean inner product on T X.
u
u
u
LOVELY PROFESSIONAL UNIVERSITY 297
