Ivancevic_Applied-Diff-Geom

16 Applied Differential Geometry: A Modern Introductiona connection is needed (see below). In particular, the exterior calculuson a totally antisymmetric tensor bundle allows for a generalization of theclassical gradient, divergence and curl operators.A jet bundle is a generalization of both the tangent bundle and thecotangent bundle. The Jet bundle is a certain construction which makes anew smooth fiber bundle out of a given smooth fiber bundle. It makes itpossible to write differential equations on sections of a fiber bundle in an invariantform. In contrast with Riemannian manifolds and their (co)tangentbundles, a connection is a tensor on the jet bundle.1.1.5 Riemannian Manifolds: Configuration Spaces for LagrangianMechanicsTo measure distances and angles on manifolds, the manifold must be Riemannian.A Riemannian manifold is an analytic manifold in which eachtangent space is equipped with an inner product g = 〈·, ·〉, in a mannerconcepts of scalar, vector and linear operator in a way that is independent of any chosenframe of reference. While tensors can be represented by multi-dimensional arraysof components, the point of having a tensor theory is to explain the further implicationsof saying that a quantity is a tensor, beyond that specifying it requires a numberof indexed components. In particular, tensors behave in special ways under coordinatetransformations. The tensor notation (also called the covariant formalism) was developedaround 1890 by Gregorio Ricci–Curbastro under the title ‘Absolute DifferentialGeometry’, and made accessible to many mathematicians by the publication of TullioLevi–Civita’s classic text ‘The Absolute Differential Calculus’ in 1900. The tensor calculusachieved broader acceptance with the introduction of Einstein’s general relativitytheory, around 1915. General Relativity is formulated completely in the language oftensors, which Einstein had learned from Levi–Civita himself with great difficulty. Buttensors are used also within other fields such as continuum mechanics (e.g., the straintensor). Note that the word ‘tensor’ is often used as a shorthand for ‘tensor–field’, whichis a tensor value defined at every point in a manifold.The so–called ‘classical approach’ views tensors as multidimensional arrays that are nDgeneralizations of scalars, 1D vectors and 2D matrices. The ‘components’ of the tensorare the indices of the array. This idea can then be further generalized to tensor–fields,where the elements of the tensor are functions, or even differentials.On the other hand, the so–called ‘modern’ or component–free approach, views tensorsinitially as abstract geometrical objects, expressing some definite type of multi–linearconcept. Their well–known properties can be derived from their definitions, as linearmaps, or more generally; and the rules for manipulations of tensors arise as an extensionof linear algebra to multilinear algebra. This treatment has largely replacedthe component–based treatment for advanced study, in the way that the more moderncomponent–free treatment of vectors replaces the traditional component–based treatmentafter the component–based treatment has been used to provide an elementarymotivation for the concept of a vector. You could say that the slogan is ‘tensors areelements of some tensor bundle’.