Linear Algebra Essentials to Differential Geometry

This is my first notes w.r.t. differential geometry study, from the book “first steps in differential geometry”, This is only the linear algebra part. Noted: this book is pretty succint in Linear Algebra part, so I’ll try to understand it by my own.

Dual of a vector space, forms and pullbacks

This section corresponds to chapter 2.8.

Dual of a vector space, aka. one-form, covector

Given V as a vector, its corresponding dual form is a map from V to R: $V^* = V\to R$. Further, \(V^*\) could be regarded as a dot-product. Then, given $v\in R^2, T\in R^{2*}, T\circ v \in R$. We could think of T as another $R^2$ vector, and the computation is dot-product. [This is supported by Theorem 2.9.22 later]

Connection between one-form $T\in V^*$ and vector space $V$

• Let $B={e_1,…e_n}$ be a basis for V, then the basis for \(V^*\) is \(B^*=\{ \epsilon_1,...,\epsilon_n \}\), s.t.
  • $\epsilon_i (e_i)=1$ and $\epsilon_i (e_j)=0$ for $j\neq i$
• Then, $T=\sum_i T(e_i)\epsilon_i$
• Intuition is very easy: let $\epsilon_i = e_i$, and they’re all orthonormal basis, while treating one-form mapping as dot-product.

Pullbacks

Projection ($\Phi$) is a map from v to w, could be regarded as a matrix $\in R^{n\times m}$, while $v\in R^n, w\in R^m$. Then pullback($\Phi^\star$) is a map from $w^\star$ back to $v^\star$, while they’re all covectors. A pullback could be regarded as the transpose of the corresponding projection matrix, namely, $\Phi^\star = \Phi^T$.

Pullback is defined this way: \((\Phi^*(T))(v) = T(\Phi(v))\), where \(T\in W^*\) and \(v\in V\). In the math understanding:

• left: pullback \(T\in w^*\) to \(\Phi(T)\in v^*\), then apply “dot-product” to v.
• right: project $v\in V$ to $w\in W$, then apply \(w^*\in W^*\) to it, and get a real value.

While in the matrix understanding:

• Left = \((\Phi^*\circ T)^T \circ v\)

• right=$T^T\circ ( \Phi\circ v )$

We could easily observe that they are equal. Noted: every time we apply dot-product, we need to transpose the former vector.

Then, we can understand \((\Phi_2\circ \Phi_1)^* = \Phi_1^*\circ \Phi_2^*\), by transpose of matrix multiplication.

Two-forms (aka. Bilinear Form) and Its Pullbacks

Since one-form (\(V\to R\)) could be regarded as dot-product (with a vector), two-form ($b \in V\times V \to R$) could be regards as a bilinear operation (with a matrix): $b(v,w) = w^TBv$, and $B$ is its corresponding matrix.

Pullbacks of bilinear forms is of type $(W\times W \to R) \to (V\times V \to R)$ , and defined by:
\((T^*B)(v_1,v_2) = B(T(v_1), T(v_2))\) where $T\in V\to W$ and $B\in W\times W \to R$.

Inner product space

Inner product space is just the normal/standard space, where norm, angle are defined. Also, the orthonormal basis of a inner product space $(V,G)$ is the same as that of $V$, where $V$ is vector space, and $G$ is a bilinear form ($V\times V\to G$).

Linear Symplectic Forms

Linear symplectic form, also known as alternating bilinear form, has a different property with bilinear form:

• Skew-symmetric: for all $v,w\in V$, $\omega(w,v) = -\omega(v,w)$

It has an intuition of cross-product.