Let $f_0, f_1:M\longrightarrow N$ be two smooth maps. Then we have induced maps at the level of forms $f^*_0, f^*_1:\Omega^k(N)\longrightarrow \Omega^k(M)$ where $\Omega^k(M)$ is the space of smooth differential forms on $M$.

Suppose $h:\Omega^k(N)\longrightarrow \Omega^{k-1}(M)$ is a chain homotopy between $f^*_0$ and $f^*_1$, that is: $$d\circ h-h\circ d=f^*_0-f^*_1,$$ where $d$ stands for the De Rham differential. Is it true that $h$ induces a vector bundle map $$H:TM\times TI\longrightarrow TN$$ such that $$\left\{\begin{array}{l} H\circ \jmath_0=df_0\\ H\circ \jmath_1=df_1 \end{array}\right.$$ where $\jmath_0, \jmath_1:TM\longrightarrow TM\times TI$ are $$\jmath_0(v):=(v, 0^{TI}(0))\quad \textrm{and}\quad \jmath_1(v):=(v, 0^{TI}(1)),$$ and $0^{TI}:I\longrightarrow TI$ is the zero section.

  • $\begingroup$ You probably mean $H \circ j_i = df_i$, right? $\endgroup$ – levap Mar 25 '17 at 21:41
  • $\begingroup$ Right, just fixed it, thanks =) $\endgroup$ – PtF Mar 25 '17 at 21:59

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