By Masahisa Adachi

This booklet covers basic thoughts within the concept of $C^{\infty }$-imbeddings and $C^{\infty }$-immersions, emphasizing transparent intuitive knowing and containing many figures and diagrams. Adachi begins with an creation to the paintings of Whitney and of Haefliger on $C^{\infty }$-imbeddings and $C^{\infty }$-manifolds. The Smale-Hirsch theorem is gifted as a generalization of the category of $C^{\infty }$-imbeddings through isotopy and is prolonged by way of Gromov's paintings at the topic, together with Gromov's convex integration concept. ultimately, as an program of Gromov's paintings, the writer introduces Haefliger's type theorem of foliations on open manifolds. additionally defined here's the Adachi's paintings with Landweber at the integrability of just about advanced buildings on open manifolds. This ebook will be a good textual content for upper-division undergraduate or graduate classes.

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Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in· square also.

The steps are exactly the same as shown under (I) and (2) of the last note, with v instead of ", except only in the lines "x ,... 2X" and "2X '"' x" which are unaltered, while, in the references, x. 13, 16 take the place of x. 12, IS respectively. [LEMMA. Since it has been proved that straight lines commensurable in length are always commensurable in square also, while those commensurable in square are not always commensurable in length also, but can of course be either commensurable or incommensurable in length, it is manifest that, if any straight line be commensurable in length with a given rational straight line, it is called rational and commensurable with the other not only in length but in square also, since straight lines commensurable in length are always commensurable in· square also.

II] But the square on CD is commensurable with the square on BF, for the straight lines are rational in square; and the rectangle DC, CB is commensurable with the rectangle FE, EG, for they are equal to the square on A ; therefore the square on CD is also incommensurable with the rectangle DC, CB. [x. r3] But, as the square on CD is to the. rectangle DC, CB, so is DC to CB; [Lemma] therefore DC is incommensurable in length with CB. [x. II] Therefore CD is rational and incommensurable in length with CB.