Text. 9. Differential Geometry

Many new and extensive fields of math investigation were opened up in the seventeenth century, making that era an outstandingly productive one in the development of maths. Unquestionably, the most remarkable math achievement of the period was the invention of the calculus by Isaac Newton and Gottfried Wilhelm von. Leibnitz. A fair share of its remarkable applicability lies in the field of geometry and there is an exceedingly vast body of geometry wherein one studies properties of curves and surfaces, and their generalizations, by means of the calculus. This body of geometry is known as "differential geometry". For the most part, differential geometry investigates curves and surfaces only in the immediate neighborhood of any one of their points. This aspect of differential geometry is known as "local differential geometry" or "differential geometry in the small". However, sometimes properties of the total structure of a geometric figure are implied by certain local properties of the figure that hold at every point of the’ figure. This leads to what is known as "integral geometry" or "global differential geometry", or "differential geometry in the large".! It is probably quite correct to say that differential geometry, at least in its modem dress, started in the early part of the eighteenth century with the interapplications of the calculus and analytic geometry. Karl Friedrich Gauss (1777-1855) introduced the fruitful method of studying the differential geometry of curves and surfaces by means of parametric representation of these objects. Bernhard Riemann introduced an improved notation and a procedure independent of any particular coordinate system employed. The tensor calculus was accordingly devised and developed. Here we find an assertion of the tendency of maths in recent times to strive for the greatest possible generalization.

Generalized differential geometries, known as Riemannian geometries were explored intensively, and this in turn led to non-Riemannian, and other, geometries. Much of this material finds significant application in relativity theory and other parts of modem physics.

Vocabulary

1. Give the main points of the text.

2. Find answer to the following questions:

1) How can you characterize the 17^th century?

2) What did it give rise to?

3) Where did the invention of calculus find use?

4) What does differential geometry investigate?

5) In what way does modern geometry differ from the early geometry?

6) How did different scientists contribute to the development of geometry?

3. Find the missing words in the text:

1) Differential geometry ___ curves and surfaces.

2) Sometimes properties of the total structure are ___ by certain local properties.

3) He introduced the ___ method of studying.

4) The tensor ___ was accordingly devised and developed.

5) There is the tendency of maths to ___ for more generalization.

6) Other geometries find ___ application in relativity theory.