On the diffetential of a mapping 2

In the case {\psi : \mathbb{R}^n \rightarrow \mathbb{R}^m} case, there is a linear map, which is “linear approximation” of {\psi}. In the manifold case, there is a similar linear map, but now it acts between tangent spaces. If {M} and {N} are smooth manifolds and {\psi \colon M \rightarrow N} is a smooth map then for each {m \in M}, the map

\displaystyle d\psi \colon T_mM \rightarrow T_{\psi(m)}N

is defined by

\displaystyle d\psi(v)(f) = v(f \circ \psi)

is called the pushforward. Actually,

\displaystyle d\psi \colon TM \rightarrow TN.

Suppose that {\dim{M} \ge \dim{N}} and {f \colon M \rightarrow N} is a differentiable mapping. We have

Definition 1 The mapping {f} is called a trivial fibration (differentiable) on {N} if there exists a differential manifold {F}, is called fibre of {f}, and a diffeomorphism

\displaystyle \phi \colon M \rightarrow N \times F

such that the following diagram is commutative


On the differential of a mapping

From on Warner’s book.

Smooth curve on manifold {M}:

A {C^\infty} mapping {\alpha : (a, b) \rightarrow M}. Let {t \in (a, b)}, we define the tangent vector of the curve {\alpha} at {t} is the vector

\displaystyle d\alpha\bigg(\dfrac{d}{dt}\bigg|_{t = 0}\bigg) \in T_{\alpha(0)}M.

we apply the formula

\displaystyle d\psi(v)(g) = v(g \circ \psi),

where {g} is an any function on {M}.

Put {\psi = \alpha(t)} and {v = \dfrac{d}{dt}\alpha(t)\big|_{t = 0}}, the above formula implies

\displaystyle d\alpha(\frac{d}{dt}\big|_{t=0})(f) = (\frac{d}{dt}\big|_{t=0})(f \circ \alpha) = \frac{d}{dt}(f \circ \alpha)\big|_{t=0}.

This is directional derivative.

Exercise of diff topo 1

BT của Munkres,

BT1, Cho M là một đa tạp chiều m có biên khác rỗng. Gọi M_0 = M \times 0M_1 = M \times 1 là hai mảnh của M. “Double” của M, ký hiệu là D(M), là không gian tô pô tạo bởi M_0 \cup M_1 bằng cách gắn đồng nhất (x,0) (x,1) với mỗi x \in Bd(M) (biên của M). Chứng minh rằng D(M) là một đa tạp không có biên m chiều.