By Keti Tenenblat

The interplay among differential geometry and partial differential equations has been studied because the final century. This courting is predicated at the indisputable fact that lots of the neighborhood houses of manifolds are expressed by way of partial differential equations. The correspondence among definite periods of manifolds and the linked differential equations should be worthy in methods. From our wisdom concerning the geometry of the manifolds we will be able to receive strategies to the equations. specifically it is very important learn differences of manifolds which guard a geometrical estate, because the analytic interpretation of those differences will offer mappings among the corresponding differential equations. Conversely, we will be able to receive geometric houses of the manifolds or perhaps turn out the non lifestyles of definite geometric constructions on manifolds from our wisdom of the differential equation. this sort of interplay among differential geometry and differential equations is the final subject matter of the e-book. the writer makes a speciality of the function performed through differential geometry within the learn of differential equations, combining the geometric and analytic elements of the idea, not just within the classical examples but additionally in effects got on the grounds that 1980, on integrable structures with an arbitrary variety of self sustaining variables. The e-book may be of curiosity to graduate scholars, researchers and mathematicians operating in differential geometry, differential equations and mathematical physics.

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**Transformations of Manifolds and Application to Differential Equations**

The interplay among differential geometry and partial differential equations has been studied because the final century. This courting relies at the undeniable fact that many of the neighborhood homes of manifolds are expressed by way of partial differential equations. The correspondence among yes sessions of manifolds and the linked differential equations should be important in methods.

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X~) there exist a. , xg. 1). 3), which does not cor:espond to any surface in H:i. 2 Theorem. (Superposition formula for the sine-Gordon equation) Let 'I/J1 and 'ljJ2 be two solutions of the sine-Gordon equation associated to 'ljJ by BT( 81 ) and BT( 82 ) respetively. Then, if 81 =F 92 , there exists a unique solution '1/J .. 5. Backlund transformation. Superposition formula 45 associated to 1/J1 and 1/J2 by BT(02 ) and BT(0 1) respetively. __ = sin( ~ 92 9 11 sin( (5 . 2 ) 4 ; ') Proof. By hypothesis 'ljJ 1 and '1/; 2 are associated to '1/; by BT(01 ) and BT(02 ) respectively, therefore '1/;l,rt 'I/J1,x2 and 1•2,, 1 •'f' 1/J2,x2 o.

Such a congruence will be called a geodesic congruence between M and M'. In the remainder of this section, we will consider a special kind of geodesic congruence which exists between parallel surfaces. Other congruences will be treated in the following section. 1 Example. e surface and let e3 be a unit normal field on M. e. 1) =H c 3 L4 where a ::/: 0 is a real constant. We say that M and M' are parallel s·urfaces at a distance a. 1), define a parallel surface? We denote by Hand K the mean and Gaussia:J.

X~,x2) f 0. e. e. ,. + (1; 2 kdx2) Zx2 + (1;• 1; hdx1 2 kdx2) z = 0. 3) for the geometrical interpretation of the Laplace transformation. Observation 1 will be useful to relate the geometrical construction with the Laplace method. Observation 4. 2). tion of the Laplace transformation. Consider a parametrized surface X(x~ox 2 ) in R3 or R4 such that be coordinate curves form a conjugate net. e. the mixed second order derivative of X is tangent to the surface. X2 = r~2x,,. 4) where fb, i = 1, 2 are the Christoffel symbols of the surface X.