Not all the geometrical structures are "equal". It would seem that the riemannian and complex structures, with their contacts with other fields of mathematics and with their richness in results, should occupy a central position in differential geometry. A unifying idea is the notion of a G-structure, which is the modern version of an equivalence problem first emphasized and exploited in its various special cases by Elie Cartan.
Shiing-Shen ChernRelated topics
central complex differential equal equivalence geometry idea mathematics modern notion position problem richness should special unifying version fields riemannianRelated quotes
Geometry is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire readiness in making plans for buildings in their grounds, and rightly apply the square, the level, and the plummet. By means of optics the light in buildings can be drawn from fixed quarters of the sky. Difficult questions involving symmetry are solved by means of geometrical theories and methods.
Vitruvius
Pasteur originally conceived the idea of germs and of destroying them. Although this started as a personal thing, it has mushroomed into DDT, killing beetles and worms, resulting in food contamination, much sickness, and trouble. Although he is regarded as hero by modern medicine, Pasteur will be treated in much the same way as a warmonger when he is judged in the spiritual world.
Michio Kushi
Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity.
Morris Kline
It is to the director of workshops and factories that it is suitable to make, by means of geometry and applied mechanics, a special study of all the ways to economize the efforts of workers... For a man to be a director of others, manual work has only a secondary importance; it is his intellectual ability (force intellectuelle) that must put him in the top position, and it is in instruction such as that of the Conservatory of the Arts and Professions, that he must develop it.
Charles Dupin
It is a far cry from the abacus to the modern keyboard accounting machine. It will be an equal step to the arithmetical machine of the future. But even this new machine will not take the scientist where he needs to go. Relief must be secured from laborious detailed manipulation of higher mathematics as well, if the users of it are to free their brains for something more than repetitive detailed transformations in accordance with established rules.
Vannevar Bush
