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mathwonk

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Since this line of thought has proved so wildly popular I will push it further:

Why would one take the previous point of view? The answer may be in the fact that many geometric surfaces of interest are not manifolds, i.e. are not actually smooth objects which are locally parametrizable by euclidean space, hence local coordinates are not available.

I.e. in that event, even the familiar tangent and cotangent bundles are actually not products. In fact one definition of a manifold, after defining the intrinsic cotangent bundle, is that the (possibly singular) variety is a manifold if and only if the cotangent bundle is locally a product.

This means one cannot define even tangent and cotangent vectors in the simple intuitive way that has been used for manifolds. I.e. we think usually first of "what is a vector at a point"? Then we think of a field of vectors on an open coordinate set, and finally we introduce changes of coordinates for different but overlapping open sets.

Then for tensors, we use the same procedure, passing from pointwise, to local, to global. I.e. we go back to a point and define a tensor at a point, then take products of that eucldiean tensor space with an open coordinate set, and finally ask how the coefficients or components of the tensor change as we change coordinates.

But what if there is no possibility of introducing local coordinates near a certain "bad" point? i.e. a singular point? such as near the origin of a cone. Then the solution in algebraic and analytic geometry is to do the local version first, (taking account of the fact that "local" does not mean "locally trivial"), and after that the pointwise version. And tensor products play a crucial role, even in the definiton of vectors and covectors.

Here is what comes out:

From a certain strange point of view, the tangent bundle to a manifold X, is the same as the normal bundle to the diagonal in the product XxX. (This is familiar in differential topology, where the euler characteristic of a manifold is defined sometimes as the self intersection of the diagonal, i.e. via Hopf's theorem, as the number of zeroes of a general vector field.) Thus the conormal bundle to the diagonal is the cotangent bundle to X.

Now the advantage of this point of view, since we have not defined cotangents or tangents yet, is that conormal bundles are more basic than tangent bundles! I.e. the conormal bundle to the diagonal in XxX, is just the family of functions vanishing on the diagonal, modulo those vanishing twice!

To see what this has to do with derivatives, note the usual difference quotient defining a derivative, i.e. [f(x) - f(a)]/(x-a). See there, that denominator is a function of two variables, a and x, hence a function on the product XxX. Moreover note that when x=a, the numerator is zero, so "deltaf" is a function on the product XxX which vanishes on the diagonal.

However as a derivative, or a differential, df is not consiered zero unless it vanishes twice on the diagonal, i.e. unless after dividing out by the first order zero, i.e. by x-a, we still get zero. Now in algebra we just divide by x-a straightaway, and we can define the derivative of f at a, as the value of the actual algebraic quotient

[f(x)-f(a)]/(x-a), at a. That is how fermat and descartes took derivatives, or found tangents. But in analysis we must take a limit to evaluate this, the usual Newton definition of the derivative.

So to sum up, the cotangent bundle of X is by definition, locally the quotient of the ideal I of functions on XxX which vanish on the diagonal, modulo I^2, those vanishing twice.

If we let C(X) be the ring of functions on X, then it turns out that the ring of functions on XxX is locally C(X)[tensor]C(X), and I is the kernel of the multiplication map C(X)[tensor]C(X)-->C(X). Then the cotangent bundle of X is locally I/I^2. This ism true even at singular, i.e. non manifold, points.

Now this is all more or less true in algebraic and analytic geometry (plus or minus my inherent inaccuracy and ignorance), but I have not checked it for C infinity functions, as my notation suggests here. maybe Hurkyl would be interested in trying this out along with his investigation of general schemes and their smooth analogues.

To pass back to the point wise situation, one defines the pointwise cotangent space as the (pointwise) localization of the module I/I^2, at the point p, and this is done, guess what? by tensoring I/I^2 with the field of coinstants at the point.

I.e. T*(p) = (I/I^2)[tensor(C(X))]R, where R say is the field of real numbers, (and the tacitly assumed homomorphism from C(X) to R, is just evaluation at p).

So tensors have a huge variety of uses.

peace,

I hope this does not kill the interest of this thread for good. Just ignore what I said here if you like.

Why would one take the previous point of view? The answer may be in the fact that many geometric surfaces of interest are not manifolds, i.e. are not actually smooth objects which are locally parametrizable by euclidean space, hence local coordinates are not available.

I.e. in that event, even the familiar tangent and cotangent bundles are actually not products. In fact one definition of a manifold, after defining the intrinsic cotangent bundle, is that the (possibly singular) variety is a manifold if and only if the cotangent bundle is locally a product.

This means one cannot define even tangent and cotangent vectors in the simple intuitive way that has been used for manifolds. I.e. we think usually first of "what is a vector at a point"? Then we think of a field of vectors on an open coordinate set, and finally we introduce changes of coordinates for different but overlapping open sets.

Then for tensors, we use the same procedure, passing from pointwise, to local, to global. I.e. we go back to a point and define a tensor at a point, then take products of that eucldiean tensor space with an open coordinate set, and finally ask how the coefficients or components of the tensor change as we change coordinates.

But what if there is no possibility of introducing local coordinates near a certain "bad" point? i.e. a singular point? such as near the origin of a cone. Then the solution in algebraic and analytic geometry is to do the local version first, (taking account of the fact that "local" does not mean "locally trivial"), and after that the pointwise version. And tensor products play a crucial role, even in the definiton of vectors and covectors.

Here is what comes out:

From a certain strange point of view, the tangent bundle to a manifold X, is the same as the normal bundle to the diagonal in the product XxX. (This is familiar in differential topology, where the euler characteristic of a manifold is defined sometimes as the self intersection of the diagonal, i.e. via Hopf's theorem, as the number of zeroes of a general vector field.) Thus the conormal bundle to the diagonal is the cotangent bundle to X.

Now the advantage of this point of view, since we have not defined cotangents or tangents yet, is that conormal bundles are more basic than tangent bundles! I.e. the conormal bundle to the diagonal in XxX, is just the family of functions vanishing on the diagonal, modulo those vanishing twice!

To see what this has to do with derivatives, note the usual difference quotient defining a derivative, i.e. [f(x) - f(a)]/(x-a). See there, that denominator is a function of two variables, a and x, hence a function on the product XxX. Moreover note that when x=a, the numerator is zero, so "deltaf" is a function on the product XxX which vanishes on the diagonal.

However as a derivative, or a differential, df is not consiered zero unless it vanishes twice on the diagonal, i.e. unless after dividing out by the first order zero, i.e. by x-a, we still get zero. Now in algebra we just divide by x-a straightaway, and we can define the derivative of f at a, as the value of the actual algebraic quotient

[f(x)-f(a)]/(x-a), at a. That is how fermat and descartes took derivatives, or found tangents. But in analysis we must take a limit to evaluate this, the usual Newton definition of the derivative.

So to sum up, the cotangent bundle of X is by definition, locally the quotient of the ideal I of functions on XxX which vanish on the diagonal, modulo I^2, those vanishing twice.

If we let C(X) be the ring of functions on X, then it turns out that the ring of functions on XxX is locally C(X)[tensor]C(X), and I is the kernel of the multiplication map C(X)[tensor]C(X)-->C(X). Then the cotangent bundle of X is locally I/I^2. This ism true even at singular, i.e. non manifold, points.

Now this is all more or less true in algebraic and analytic geometry (plus or minus my inherent inaccuracy and ignorance), but I have not checked it for C infinity functions, as my notation suggests here. maybe Hurkyl would be interested in trying this out along with his investigation of general schemes and their smooth analogues.

To pass back to the point wise situation, one defines the pointwise cotangent space as the (pointwise) localization of the module I/I^2, at the point p, and this is done, guess what? by tensoring I/I^2 with the field of coinstants at the point.

I.e. T*(p) = (I/I^2)[tensor(C(X))]R, where R say is the field of real numbers, (and the tacitly assumed homomorphism from C(X) to R, is just evaluation at p).

So tensors have a huge variety of uses.

peace,

I hope this does not kill the interest of this thread for good. Just ignore what I said here if you like.

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