What Did People Use Before Toilet Paper?

Modern toilet paper was not commonly available in the United States until the mid 19th century. Before it was manufactured in the ubiquitous 4 ½" rolls we all know and love, toilet paper came in bundles of flat sheets, roughly the size of the box of today's facial tissues (which are larger sheets, folded).

What Did People Use Before Toilet Paper? 1

Tangent bundle of a quotient by a proper action

I think the best answer for your general question comes from observing that $pi:Xto X/G$ is a surjective submersion. Hence for each $xin X$, you can identify the tangent space $T_pi(x)X/G$ with the quotient of $T_xX$ by the tangent space of the orbit through $x$. The latter is the subspace of $T_xX$ spanned by the fundamental vector fields for the action associated to elements in the Lie algebra $mathfrak g$ of $G$. Here for $Ainmathfrak g$, the fundamental vecor field is defined by $zeta_A(x)=fracddt|_t=0xcdot exp(tA)$. For $mathbb CP^n$ realized as a quotient of $S^2n1$, this gives you and identification of the tangent space to the complex line spanned by $x$ as the quotient of $x^perp$ (real orthocomplement) by imaginary multiples of $x$. This is the right space, but not quite the identification that you would like to get. To get an identification like the one you want, you probably have to realize $mathbb CP^n$ as a homogeneous space (since this allows you to compare tangent spaces at different points to some extent). The most general version of this is viewing it as a homogeneous space of $GL(n,mathbb C)$. Then for each line $DsubsetBbb C^n1$, the map $Amapsto A(D)$ defines a surjective submersion from $GL(n,Bbb C)$ onto $mathbb CP^n$. Hence you get an identification of $T_Dmathbb CP^n$ with the quotient of the tangent space of $GL(n,mathbb C)$ (which is just the space of complex $ntimes n$-matrices) by the Lie algebra of the stabilizer of $D$, which is simply formed by all matrices mapping $D$ to itself. This quotient can be identified with the space of linear maps from $D$ to $mathbb C^n1/D$, and this is the identification you want. (If you prefer to involve a complex orthocomplement, you can set up a similar picture with $U(n)$ or $SU(n)$ acting on the space of lines. ).

Where to buy bundle clothes ?

Bundle Of Clothes

What Did People Use Before Toilet Paper? 2

Introduction a good text on principal bundle

My personal favorites are "Topology, Geometry, & Gauge Fields: Foundations" by Gregory Naber and "Gauge Theory & Variational Principles" by David Bleecker. Naber's book does a lot of in-depth calculations and makes a solid attempt at explaining the motivation behind the various mathematical notions introduced, two things that the authors of more "sophisticated" books will not debase themselves with. Bleecker's book is nice because it's a cheap, compact, well-written paperback. It's perfunctory in places, but also goes deeper, especially with mathematical topics related to physics. I will also add a suggestion that I have no personal experience with, which I just discovered by browsing Amazon: "Principal Bundles: The Classical Case" by Stephen Bruce Sontz. A look at the table of contents was enough to pique my interest, so I include it here for what it's worth.

Which PS3 bundle is better?

just buy the basic package. and get the cheapest HDMI cable at HMV ($12.99). That cheap HDMI cables works the same as my Sony HDMI cable. Pain is not a good game. It's so boring. And get a logitech wireless controller with rumble ($35.99 @ tigerdirect.ca)) for PS3 as your second controller.. it's amazing.. mise is actually a wirless controller for PS2 and I just bougth a $10 adapter but the rumble still works and the response is awesome! The battery lasts for 3 months. So for the 80GB, if you get just the console@ $399 HDMI cable @ $12.99 (from HMV or $9.99 from Real Canadian Superstore) extra controller @ $35.99 1 game @ say $60... total is only...$507.98 (before tax). You save $100 bucks already for maybe 2 more games...

Tangent bundle equivalence not a pushforward

Given any smooth map $f: Bbb R^n to GL_n(Bbb R)$, you can construct a bundle map $T_f: TBbb R^n to TBbb R^n$ by $T_f(x,v) = (x,f(x)v)$. This is not induced by a diffeomorphism, because it lies over the identity! This construction (roughly) proves that bundle maps induced by diffeomorphisms are of infinite codimension in the space of all bundle automorphisms.Two points are worth bringing up. First, this is related to the fact that, given any two Riemannian structures $g_1$ and $g_2$ on a vector bundle $E$, there is a bundle isomorphism $f: E to E$ that takes $g_1$ to $g_2$. (That is, 'all Riemannian structures on a vector bundle are isomorphic.') But it is far from true that any two diffeomorphic Riemannian manifolds are isometric!Second, the desire to take a random bundle map and promote it to one induced by a smooth map is clearly quite desirable when doing differential topology or geometry - you can take algebraic-topological information (the space of bundle maps) to geometric structure (the space of embeddings or what have you). This idea is usually known as the h-principle. One famous example is the Smale-Hirsch immersion theory, which states that the space of bundle maps $TM to TN$ deformation retracts onto the space of immersions $M looparrowright N$. (Corollary: sphere eversion.)

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