How to Spot a Tensor
A bilinear form is a map which takes two vectors and maps them to a
real number and is linear in each of them. In two dimensions, the
general form is
B(A,B)=B((a1a2),
(b1b2))
=ma1b1+na1b2+pa2b1+qa2b2
=(a1a2)(mnpq)(b1,b2).
Any inner product can be written as a bilinear form. The standard
cartesian inner product corresponds to the identity matrix. When we
use the first fundamental form, I(A,B)
to find the cosine of the angle between the tangent vectors A and B, we are looking at (a1a2)(g11g12g21g22)(b1,b2).
A tensor is a mathematical object (yes, that's vague) that behaves a
certain way under a change of coordinates (yes, that's vague, also). Different kinds of tensors
transform different ways, but they are all defined in terms of the
Jacobian matrix between the two coordinate systems. (If the two sets
of coordinates are (u,v) and (w,z), the matrix is
J(u,v,w,z)=(dudwdudzdvdwdvdz)
- see the third semester calculus labs.)
This is because tensors live in the tangent space defined by Xu and Xv.
The first and second fundamental forms are tensors of type (0,2), which means I(u,v)=JI(v,w)JT and I(u,v)=JII(v,w)JT. The Weingarten map is a
tensor of type (1,1) because L(u,v)=JL(v,w)J-1. The indices are based on how
many times and in what form the Jacobian appears in the
transformation. A scalar is a tensor of type
(0,0), meaning that it has nothing to do with which basis you use.
You may see the second fundamental form written as:
L11du⊗du+L12du⊗dv+L21dv⊗du+L22dv⊗dv.
The operator ⊗ is called a tensor product. For most purposes this means
the same thing as what we wrote earlier. The difference is that here, we have
a form which is waiting for two vectors as input. The tensor product applied to the vectors A and B is defined w⊗v·(A,B)=(w·A)(v·B)
where the multiplication on the right is just scalar multiplication.
In the context of the above expression, du is a
differential form. This is a map which projects from the tangent space to the
reals. It takes a tangent vector to its Xu component; dv is defined analogously. If we apply this tensor product of differential
forms to two vectors, A and B, we get
L11du⊗du·(A,B)+L12du⊗dv·(A,B)
+L21dv⊗du·(A,B)+L22dv⊗dv·(A,B).
,
we see that it is the same thing the bilinear form, (a1a2)(L11L12L21L22)(b1,b2) in the Xu, Xv basis.
Note that II is rarely used as a bilinear
form. It is almost always used as a quadratic form, i.e. II(X',X').