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 X_u and X_v. The first and second fundamental forms are tensors of type (0,2), which means I(u,v)=JI(v,w)J^T and I(u,v)=JII(v,w)J^T. 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 X_u 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 X_u, X_v 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^').