Which Book on Riemannian Geometry Balances Intuition and Minimal Prerequisites?

[mathwonk]

abstract BS on manifolds

Now to do all this "intrinsically", without the coordinates of 3 space, someone had the brilliant but very cumbersome idea to equip the surface with a different coordinate system at every point, e.g. a tangent plane at each point, and a projection from each tangent plane down onto the surface, at least locally.

Then to measure lengths, angles, and distances, one needs a dot product, but one needs a different one in every tangehnt space, i.e. one needs a dot product at every point. To make it less explanatory to the uninitiated, and claim the blessing of deities, we call this choice a "Riemannian metric".

Then we have everything you have in euclidean space, but we have a different one at each point! we have families of diferential forms, one form, 2 forms, etc... we correctly recognize these as "sections" of various "bundles" (of vector and covector spaces).

so we mesmerize the innocent by telling them a differential one form is a section of the cotangent bundle, and since they already know what determinants are, we use a new word to frighten them, calling a differential k form a section of the kth exterior algebra on the cotangent bundle.

by now they are quivering in fear, and we introduce curvature in terms of "connections", or even "Koszul connections", maybe to make them think of the monsters in ghostbusters, but which is just a way of taking derivatives.

spheres are completely forgotten, we have insured that no one will any longer grasp how simple curvature is. we can rest, our work is done.

 

[quantum123]

Interesing ..
2 points:-
1) Does it extend to N dimension? Can you extend this to pseudo-riemannian?
2) Do you mean that it is better to have an ambient space for the surface to embedd and that sphere is in the ambient space? This is not very popular among the relativity people. The idea of ambient space is considered unnecssary. Intrinsic is considered beautiful.

 

[mathwonk]

2) that is the first big result of gauss, called the "theorema egregium" or "big theorem". namely that the curvature of a surface embedded in three space, and measured using the gauss map to the sphere, is exactly the same as the intrinsic version of the curvature, defined using only the riemannian metric induced by the embedding. Or in plain english "the metric determines the curvature". so this intrinsic point of view is due to gauss and riemann. the current general relativity people are only following their 150 year old lead. unfortunately they often decline to use modern truly intrinsic mathematical language for these concepts, but stick with exactly the same outmoded tools as used by riemann and einstein.

the purpose of spivaks first two volumes is to teach in vol 1 the modern language of bundels etc, then show in vol2 how they enablle one to better understand and appreciate the intrinsic conceprs riemann was trying to express without modern terminology available. inded the importance of his ideas was one thing that led people to try to express it more naturally.

1) and yes riemann showed how to generalize gauss' notion of curvature of two dimensional surfaces, to a notion of curvature in n dimensions, essentially by acting on all 2 dimensional surfaces in the larger manifold. that is riemanns big contribution,a nd the one which apparently caused gauss to leave his lecture exhulting on riemann's marvellous and fertile originaity.

i don't know what exactly pseudo riemannian means (probably relax the positive definiteness), but i will suggest the an swer is yes one can do something similar.

and i can say all this without knowing essentially anythign abut differential geometry, merely knowing how to look at something, and what books to look at.
read spivak if you want to understand the subject.

 

[mathwonk]

i do not mean ambiently embedded surfaces are better, just easier to understand. Note the key point is that the tangent vectors in euclidean space can all be translated to the origin, i.e,. "the tangent bundle of R^n is 'trivial'".

so all one needs for a gauss map is to be embedded in a manifold with trivial tangent bundle. and every manifold does embed in euclidean space. moreover complex and real tori also have trivial tangent bundles, so all smooth subvarieties of tori also have gauss maps, and these were used crucially in the study of curves in their jacobians, by andreotti and mayer.

Even in manifolds without trivial tangent bundles, gauss maps, i.e. maps induced on tangent bundles, i.e. derivatives, have been used in study of moduli spaces and maps between them, especially by Carlson and Griffiths, and (earlier but less clearly) in my thesis.

 

[cliowa]

Getting back to the title of this thread: I heard a lot of good stuff about Bishop and Crittendens https://www.amazon.com/dp/0821829238/?tag=pfamazon01-20.

Has anybody else heard of it?

 

[mathwonk]

it is a very famous old book, with excellent content, but i have never heard it compared with the ones recommended here in terms of readability. but if you can read it, it will serve you well.