The Product Metric Map to Each Coordinate Function Continuous

Coordinate Map

it made sure that coordinate maps were continuous and it secured the existence of sufficiently many "neighbourhoods" to constitute a neighbourhood basis (from our point of view).

From: History of Topology , 1999

Sample Design

W. Penn Handwerker , in Encyclopedia of Social Measurement, 2005

Nonprobability Samples

If you select a predetermined number or proportion of cases with specific case characteristics, or from specific enumeration units, transects, or sets of map coordinates, you produce a quota sample. If you select cases on the basis of case characteristics to acquire specific forms of information, you produce a purposive (judgment) sample. If you select cases simply because they will participate in your study, you produce an availability (convenience) sample. If cases become available because one case puts you in contact with another, or other cases, you produce a snowball sample.

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The Concept of Manifold, 1850–1950

Erhard Scholz , in History of Topology, 1999

5.4. Finally the "modern" axiomatic concept

There was, of course, still another line of research, more closely linked to differential geometry, where manifolds played an essential role, and purely topological aspects (independently of whether continuous, combinatorial, or homological ones) did not suffice and still needed elaboration. In North America Oswald Veblen and his students formed an active center in both fields of topology and modern geometry. Veblen and his student J.H.C. Whitehead, coming from (and going back to) Oxford, brought the axiomatization of the manifold concept to a stage which stood up to the standards of modern mathematics in the sense of the 20th century (Veblen and Whitehead, 1931, 1932). Veblen was an admirer of the Göttingen tradition of mathematics, in particular, F. Klein and D. Hilbert, and cooperated closely with H. Weyl, the broadest representative of his own generation from the Klein and Hilbert tradition. Veblen and J.H.C. Whitehead combined a view of the central importance of structure groups for geometry (generalizing the Erlanger program) with Hilbert's embryonic characterization of manifolds by coordinate systems; and they took care that the topologization of the underlying set would satisfy Hausdorff's axioms for a topological space.

They characterized the structure of a manifold by the specification of a regular groupoid G ("pseudogroup") of transformations of open sets ("regions") in ℝ ⁿ , allowing as main examples C ⁱ -transformations of open sets (i = 0,…, ∞, or i = ω). The n-dimensional manifold of structure G in the sense of Veblen and Whitehead consists in a set M and a system of admissible coordinate systems φ: U → V with bijective maps φ onto regions V ⊂ ℝ ⁿ , defined for U ∈ U ⊂ P(M), such that three groups of axioms hold:

(A)

Basic axioms for admissible coordinate systems:

Changes of coordinates are given by maps from the structure groupoid G and each coordinate map may be changed by a transformation from G (axioms A ₁, A ₂). Moreover, to each coordinate map φ: U → V a restriction to U' ⊂ V such that φ(U') = V' is an n-cell V' in ℝ ⁿ is also an admissible coordinate system (A ₃). U' is called an n-cell in the manifold.

(B)

Union of compatible coordinate systems:

If for a collection of admissible coordinate systems φ: U _i → V _i (i ∈ I), with n-cells as coordinate images V _i , the coordinate maps coincide on overlaps $(U_i\cap U_i\ne \phantom{0}/)$ , then the "union" of coordinate systems defined in the obvious way, φ: ∪ _i U _i → ∪ _i V _i , is also admissible (axiom B ₁). Each admissible coordinate system can be represented as such a union (B ₂). ¹⁰¹

(C)

Topological axioms:

For intersecting n-cells U, U' in M with p ∈ U ∩ U' there is an n-cell U″ ⊂ U ∩ U' containing p (axiom C ₁). For each two different points p, q ∈ M there exist nonintersecting coordinate neighbourhoods U _p , U _q of p and q, respectively (C ₂). Finally, M contains at least two different points (C ₃).

Taking n-cells in M, containing p, as neighbourhoods of p the axioms of Veblen and Whitehead give a structure of a Hausdorff space on M (without second countability axiom) (Veblen and Whitehead, 1931, p. 95; 1932, p. 79).

Whitehead and Veblen presented their axiomatic characterization of manifolds of class G first in a research article in the Annals of Mathematics (Veblen and Whitehead, 1931) and in the final form in their tract on the Foundations of Differential Geometry (Veblen and Whitehead, 1932). Their book contributed effectively to a conceptual standardization of modern differential geometry, including not only the basic concepts of continuous and differentiable manifolds of different classes, but also the "modern" reconstruction of the differentials dx = (dx ₁,…, dx _n ) as objects in tangent spaces to M. ¹⁰² Basic concepts like Riemannian metric, affine connection, holonomy group, covering manifolds, etc. followed in a formal and symbolic precision that even from the strict logical standards of the 1930-s there remained no doubt about the wellfoundedness of differential geometry in manifolds. Moreover they made the whole subject conceptually accessible to anybody acquainted with the language and symbolic practices of modern mathematics.

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An Introduction to Homological Algebra

In Pure and Applied Mathematics, 1979

Second Proof

By Lemma 4.45, the natural map π:A → A/J A is an essential epimorphism. Now A/J A is a f.g. R/J-module, i.e., a finite-dimensional left vector space over the division ring R/J; hence A/J ≅ (R/J) ⁿ , the sum of n copies of R/J. Consider the diagram

where φ:R ⁿ → (R/J) ⁿ is the natural map on each summand. Each coordinate map of φ is an essential epimorphism ( Lemma 4.45 applied to R itself), so Exercise 3.33 shows that φ :R ⁿ → A/J A is a projective cover. Since R ⁿ is projective (even free), there is a map ɛ:R ⁿ → A making the diagram commute: πɛ = φ. Now ker ɛ ⊂ ker φ, so ker ɛ is a superfluous submodule of R ⁿ (since ker φ is). Finally, if a ∈ A, there exists x ∈ R ⁿ with πa = φx = πɛx; thus, ɛx − a ∈ ker π = J A. Hence A = imɛ + J A, so that ɛ is epic because J A is a superfluous submodule. Therefore ɛ:R ⁿ → A is an essential epimorphism, i.e., a projective cover.

One can axiomatize this proof if one can find an ideal to play the role of J. Once we do this, we will be able to prove the existence of projective covers when R is left artinian.

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Smooth Manifolds

Steven H. Weintraub , in Differential Forms (Second Edition), 2014

4.3 Further constructions

In this section we first consider exterior differentiation and exterior products of differential forms on arbitrary smooth manifolds with boundary, and we then consider push-forwards of tangent vectors and pull-backs of differential forms under arbitrary smooth maps between smooth manifolds with boundary. Our strategy will be the same as in the last section. We will relate these to the situation of open sets in ${\mathbb{R}}^n$ , where we already know how to do things.

In general, we will see that things work "just the same as before." But we stress to the (impatient) reader that we needed to do things in ${\mathbb{R}}^n$ first before we could do them in general. Furthermore, while our constructions here are general, and the things we construct are important in themselves, in order to do concrete computations we must work in "local coordinates," i.e., we must transfer the situation to ${\mathbb{R}}^n$ by using coordinate charts and compute there.

Suppose we have an arbitrary smooth map $F:M_1⟶M_2$ between smooth manifolds $M_1$ and $M_2$ .

We first see how to push tangent vectors forward. This is the key step from which everything else follows.

Definition 4.3.1

Let $M_1$ and $M_2$ be smooth manifolds and $F:M_1⟶M_2$ be a smooth map. Let $p_1\in M_1$ and let $p_2=F(p_1)\in M_2$ . Let ${\mathbf{v}}_1$ be a tangent vector to $M_1$ at $p_1$ . Then the push-forward ${\mathbf{v}}_2=F_{\ast }({\mathbf{v}}_1)$ is the tangent vector to $M_2$ at $p_2$ defined as follows:

Let $k_1:{\mathcal{O}}_1⟶{\mathcal{U}}_1$ with $p_1\in {\mathcal{U}}_1$ and let $k_2:{\mathcal{O}}_2⟶{\mathcal{U}}_2$ with $p_2\in {\mathcal{U}}_2$ . Let $q_1=k_1^{-1}(p_1)$ and $q_2=k_2^{-1}(p_2)$ . Let ${\mathbf{v}}_1={(k_1)}_{\ast }({\mathbf{w}}_1)$ . Then ${\mathbf{v}}_2={(k_2)}_{\ast }({\mathbf{w}}_2)$ where

${\mathbf{w}}_2=G_{\ast }({\mathbf{w}}_1)=G^{\prime }(q_1)({\mathbf{w}}_1)\mathrm{where}G=k_2^{-1}\circ F\circ k_1\text{,}$

i.e.,

$F_{\ast }={(k_2)}_{\ast }\circ G_{\ast }\circ {(k_1)}_{\ast }^{-1}.◊$

Remark 4.3.2

Let us emphasize what we are doing in Definition 4.3.1. We are trying to define $F_{\ast }$ . We already know how to define $G_{\ast }$ , as $G$ is a map between open subsets of Euclidean space; indeed $G_{\ast }$ is just the derivative map $G^{\prime }$ . We also know how to define ${(k_1)}_{\ast }$ and ${(k_2)}_{\ast }$ , as $k_1$ and $k_2$ are coordinate maps. Then Definition 4.3.1 states that $F_{\ast }$ is defined precisely by requiring that the following diagram commute:

$\begin{array}{cc}\hfill & T{\mathcal{U}}_1\stackrel{F_{\ast }}{⟶}T{\mathcal{U}}_2\hfill \\ \hfill & {}_{{(k_1)}_{\ast }}\uparrow \uparrow {}_{{(k_2)}_{\ast }}\hfill \\ \hfill & T{\mathcal{O}}_1\stackrel{G_{\ast }}{⟶}T{\mathcal{O}}_2◊\hfill \end{array}$

Theorem 4.3.3

(1)

Let $M$ be a smooth manifold and let $F:M⟶M$ be the identity map. Then $F_{\ast }:\mathit{TM}⟶\mathit{TM}$ is the identity map.

(2)

Let $F_1:M_1⟶M_2$ and $F_2:M_2⟶M_3$ be smooth maps between smooth manifolds. Let $F:M_1⟶M_3$ be the composition $F=F_2\circ F_1$ . Then

$F_{\ast }={(F_2)}_{\ast }\circ {(F_1)}_{\ast }:{\mathit{TM}}_1⟶{\mathit{TM}}_3\text{.}$

Proof

(1) is obvious.

For (2), let $p_1\in M_1$ be an arbitrary point, let $p_2=F_1(p_1)\in M_2$ , and let $p_3=F_2(p_2)=F(p_1)\in M_3$ . Let $k_1:{\mathcal{O}}_1⟶{\mathcal{U}}_1\subseteq M_1$ , $k_2:{\mathcal{O}}_2⟶{\mathcal{U}}_2\subseteq M_2$ , and $k_3:{\mathcal{O}}_3⟶{\mathcal{U}}_3\subseteq M_3$ be coordinate maps, with $p_1\in {\mathcal{U}}_1\text{,}{p}_2\in {\mathcal{U}}_2$ , and $p_3\in {\mathcal{U}}_3$ . Let $q_1=k_1^{-1}(p_1)\in {\mathcal{O}}_1\text{,}{q}_2=k_2^{-1}(p_2)\in {\mathcal{O}}_2$ , and $q_3=k_3^{-1}(p_3)\in {\mathcal{O}}_3$ .

Finally, let $G_1:{\mathcal{O}}_1⟶{\mathcal{O}}_2$ be the composition $G_1=k_2^{-1}\circ F_1\circ k_1$ , let $G_2:{\mathcal{O}}_2⟶{\mathcal{O}}_3$ be the composition $G_2=k_3^{-1}\circ F_2\circ k_2$ , and let $G:{\mathcal{O}}_1⟶{\mathcal{O}}_3$ be the composition $G=G_2\circ G_1=k_3^{-1}\circ F\circ k_1$ . Then by Definition 4.3.1 and Lemma 3.2.5,

$\begin{array}{cc}\hfill F_{\ast }=& {(k_3)}_{\ast }\circ G_{\ast }\circ {(k_1)}_{\ast }^{-1}\hfill \\ \hfill =& {(k_3)}_{\ast }\circ {(G_2\circ G_1)}_{\ast }\circ {(k_1)}_{\ast }^{-1}\hfill \\ \hfill =& {(k_3)}_{\ast }\circ {(G_2)}_{\ast }\circ {(G_1)}_{\ast }\circ {(k_1)}_{\ast }^{-1}\hfill \\ \hfill =& {(k_3)}_{\ast }\circ {(G_2)}_{\ast }\circ {(k_2)}_{\ast }^{-1}\circ {(k_2)}_{\ast }\circ {(G_1)}_{\ast }\circ {(k_1)}_{\ast }^{-1}\hfill \\ \hfill =& {(F_2)}_{\ast }\circ {(F_1)}_{\ast }.\square \hfill \end{array}$

The proof of Theorem 4.3.3 appears to be just chasing symbols around. But there is one part of the proof that has real content. The heart of this proof is Lemma 3.2.5, which told us that in the situation of maps between open subsets of Euclidean space, $G_{\ast }={(G_2\circ G_1)}_{\ast }={(G_2)}_{\ast }\circ {(G_1)}_{\ast }$ . (Recall the proof of that was just the chain rule, as $G_{\ast }=G^{\prime }$ .) The rest of this proof is just transferring that equality to the more general situation of maps between smooth manifolds.

Now once we have generalized the notion of the induced map on tangent vectors from maps between regions in Euclidean space to maps between smooth manifolds, the remainder of our constructions generalizes in exactly the same way. So we can now essentially repeat some of our earlier definitions and results, with language and proofs that are virtually identical, and so we omit the proofs.

We have the following generalization of Definitions 3.3.1, 3.3.4, and 3.3.6 Definition 3.3.1 Definition 3.3.4 Definition 3.3.6 .

Definition 4.3.4

Let $F:M_1⟶M_2$ be a smooth map between smooth manifolds.

(1)

Let $f$ be a 0-form on $M_2$ , i.e., a smooth function $f:M_2⟶\mathbb{R}$ . Then the pull-back $F^{\ast }(f)$ is the 0-form on $M_1$ , i.e., the smooth function, $F^{\ast }(f)=f\circ F:M_1⟶\mathbb{R}$ .

(2)

For $j>0$ let $\phi$ be a $j$ -form on $M_2$ . Then the pull-back $F^{\ast }(\phi )$ is the $j$ -form on $M_1$ defined by $(F^{\ast }(\phi ))({\mathbf{v}}_p^1\text{,}\dots \text{,}{\mathbf{v}}_p^j)=\phi (F_{\ast }({\mathbf{v}}_p^1)\text{,}\dots \text{,}$ $F_{\ast }({\mathbf{v}}_p^j))$ . $◊$

Theorem 4.3.5

(1)

Let $M$ be a smooth manifold and let $F:M⟶M$ be the identity map. Then for any $j\text{,}{F}^{\ast }:{\mathrm{\Omega }}^j(M)⟶{\mathrm{\Omega }}^j(M)$ is the identity map.

(2)

Let $F_1:M_1⟶M_2$ and $F_2:M_2⟶M_3$ be smooth maps between smooth manifolds. Let $F:M_1⟶M_3$ be the composition $F=F_2\circ F_1$ . Then for any $j$ ,

$F^{\ast }={(F_1)}^{\ast }\circ {(F_2)}^{\ast }:{\mathrm{\Omega }}^j(M_3)⟶{\mathrm{\Omega }}^j(M_1)\text{.}$

Lemma 4.3.6

Let $F:M_1⟶M_2$ be a map between smooth manifolds. For any smooth functions $f_1$ and $f_2$ on $M_2$ , and any $j$ -forms ${\phi }_1$ and ${\phi }_2$ on $M_2$ ,

$F^{\ast }(f_1{\phi }_1+f_2{\phi }_2)=F^{\ast }(f_1)F^{\ast }({\phi }_1)+F^{\ast }(f_2)F^{\ast }({\phi }_2)\text{.}$

Now we consider some other constructions. Again we first consider them in a special case and then in general.

Definition 4.3.7

Let $M$ be a smooth manifold and let ${\phi }_1$ and ${\phi }_2$ be differential forms on $M$ . In the notation of Definition 4.2.10, let ${\psi }_1=k^{\ast }({\phi }_1{\mid }_{\mathcal{U}})$ and ${\psi }_2=k^{\ast }({\phi }_2{\mid }_{\mathcal{U}})$ . The exterior product ${\phi }_1{\phi }_2$ of ${\phi }_1$ and ${\phi }_2$ is the differential form defined by ${\psi }_1{\psi }_2=k^{\ast }({\phi }_1{\phi }_2{\mid }_{\mathcal{U}})$ . $◊$

Lemma 4.3.8

Let $k:\mathcal{O}⟶\mathcal{U}$ be a coordinate patch on $M$ . Then for any differential forms ${\phi }_1$ and ${\phi }_2$ on $\mathcal{U}$ , $k^{\ast }({\phi }_1)k^{\ast }({\phi }_2)=k^{\ast }({\phi }_1{\phi }_2)$ , i.e., exterior product commutes with pull-back by coordinate maps.

Proof

This is true because we defined the exterior product precisely by this commutativity requirement. By Definition 4.3.7: $k^{\ast }({\phi }_1)$ $k^{\ast }({\phi }_2)={\psi }_1{\psi }_2=k^{\ast }({\phi }_1{\phi }_2)$ . $\square$

Theorem 4.3.9

Let $F:M_1⟶M_2$ be a smooth map between smooth manifolds. Let ${\phi }_1$ be an $i$ -form and let ${\phi }_2$ be a $j$ -form on $M_2$ . Then

$F^{\ast }({\phi }_1{\phi }_2)=F^{\ast }({\phi }_1)F^{\ast }({\phi }_2)\text{,}$

i.e.,

$\mathit{Pull}\text{-}\mathit{back}\mathit{commutes}\mathit{with}\mathit{exterior}\mathit{product}\text{.}$

Equivalently, the following diagram commutes:

$\begin{array}{cc}\hfill & {\mathrm{\Omega }}^i(M_1)\times {\mathrm{\Omega }}^j(M_1)\stackrel{F^{\ast }\times F^{\ast }}{⟵}{\mathrm{\Omega }}^i(M_2)\times {\mathrm{\Omega }}^j(M_2)\hfill \\ \hfill & \downarrow \downarrow \hfill \\ \hfill & {\mathrm{\Omega }}^{i+j}(M_1)\stackrel{F^{\ast }}{⟵}{\mathrm{\Omega }}^{i+j}(M_2)\hfill \end{array}$

where the vertical maps are products.

Proof

This follows directly from Lemma 4.3.8, which tells us that pull-back of coordinate maps commutes with exterior product, and Theorem 3.4.1, which tells us that pull-back of smooth maps between open subsets of Euclidean space commutes with exterior product. $\square$

Definition 4.3.10

Let $M$ be a smooth manifold and let $\phi$ be a differential form on $M$ . In the notation of Definition 4.2.10, let $\psi =k^{\ast }(\phi {\mid }_{\mathcal{U}})$ . The exterior derivative $d\phi$ of $\phi$ is the differential form defined by $d\psi =k^{\ast }(d\phi {\mid }_{\mathcal{U}})$ . $◊$

Lemma 4.3.11

Let $k:\mathcal{O}⟶\mathcal{U}$ be a coordinate patch on $M$ . Then for any differential form $\phi$ on $\mathcal{U}\text{,}d(k^{\ast }(\phi ))=k^{\ast }(d\phi )$ , i.e., exterior differentiation commutes with pull-back by coordinate maps.

Proof

This is true because we defined the exterior derivative precisely by this commutativity requirement. By Definition 4.3.10: $d(k^{\ast }(\phi ))=d\psi =k^{\ast }(d\phi )$ . $\square$

Theorem 4.3.12

Let $F:M_1⟶M_2$ be a smooth map between smooth manifolds. Let $\phi$ be a $j$ -form on $M_2$ . Then

${\mathit{dF}}^{\ast }(\phi )=F^{\ast }(d\phi )\text{,}$

i.e.,

$\mathit{Pull}\text{-}\mathit{back}\mathit{commutes}\mathit{with}\mathit{exterior}\mathit{differentiation}\text{.}$

Equivalently, the following diagram commutes:

$\begin{array}{cc}\hfill & {\mathrm{\Omega }}^j(M_1)\stackrel{F^{\ast }}{⟵}{\mathrm{\Omega }}^j(M_2)\hfill \\ \hfill & d\downarrow \downarrow d\hfill \\ \hfill & {\mathrm{\Omega }}^{j+1}(M_1)\stackrel{F^{\ast }}{⟵}{\mathrm{\Omega }}^{j+1}(M_2)\hfill \end{array}$

Proof

This follows directly from Lemma 4.3.11, which tells us that pull-back of coordinate maps commutes with exterior differentiation, and Theorem 3.4.8, which tells us that pull-back of smooth maps between open subsets of Euclidean space commutes with exterior differentiation. $\square$

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Manifold Geometry

C.T.J. Dodson , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

III.A Manifolds

An n-manifold is a set M of points with the following properties:

1.

M can support the notion of continuous functions on it; (we take M to be Hausdorff with countable base).

2.

M is the union of a collection {U_α∣α∈A} of its subsets; (the collection is an open cover for M).

3.

For each α in the indexing set A there is a continuous equivalence between U _α and E ⁿ ; (homeomorphisms Φ_α: U_α  →   E ⁿ give coordinates).

4.

The change of coordinate maps are smoothly differentiable; ( ${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}$ is a diffeomorphism if U _α∩U _β  ≠   ∅).

We call the collection {(U _α, ϕ_α) α∈A} an atlas of charts for M. Properties 1–4 are clear enough but it is worth noting that 4 is meaningful because maps like Φ_α  ∘   α _β ⁻¹ go between pieces of E ⁿ on which we suppose differentiability to be well understood—namely, calculus on $\mathbb{R}$ ⁿ attached to each point. It is property 4 that enables us to say what we mean by a map between two manifolds being differentiable. Let M′ be an m-manifold with atlas {(V _λ, ψ_λ)   ∣   λ   ∈ B}. Then a map

$\text{f}:\text{M}\to {\text{M}}^{\prime }$

is called differentiable if and only if the composite maps ${\psi }_{\lambda }\circ \text{f}\circ {\phi }_{\alpha }^{-1}$ are differentiable wherever f U _α ∩ V _λ  ≠   ∅, for any α   ∈ A and λ   ∈ B. The reason for this is apparent from the diagram

Evidently, we are borrowing for manifolds the already known property of differentiability of maps from E ⁿ to E ^m . Property 4 is spoken of as giving a differentiable or smooth structure to M.

A similar trick of borrowing is employed to define T _x M, the tangent space to M at x  ∈ M. If x  ∈   U_α then we certainly have a nice vector space tangent at Φ_α(x)   ∈ E ⁿ and it is isomorphic to $\mathbb{R}$ ⁿ . The problem is that we may also have x  ∈   U_β. In that case ${\phi }_{\alpha }\circ {\phi }_{\beta }^{-1}\text{and}{\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}$ have derivatives that are actually isomorphisms between the tangent spaces at Φ_α(x) and Φ_β(x); in fact, they will appear as invertible Jacobian matrices with entries the partial derivatives, from calculus on $\mathbb{R}$ ⁿ . Such isomorphisms are used to take equivalence classes of E ⁿ -tangent spaces for synthesizing T _x M. Then the process can be rounded off nicely because the derivative of a map between manifolds actually becomes a linear map between tangent spaces.

The totality of all tangent spaces to an n-manifold M is actually itself a 2n-manifold with point set

$\text{T}\text{M}=\underset{\text{x}\in \text{M}}{\cup }{\text{T}}_{\text{x}}\text{M}=\{(\text{x},\text{u})\mid \text{x}\in \text{M},\text{u}\in {\text{T}}_{\text{x}}\text{M}\}$

and atlas {(TU _α, Tϕ_α) ∣ α   ∈ A}, where $\text{T}{\text{U}}_{\alpha }={\cup }_{\text{x}\in {\text{U}}_{\alpha }}{\text{T}}_{\text{x}}\text{M}$ and Tϕ_α is the derivative of Φ_α. We call this manifold the tangent bundle to M and it is noteworthy that it comes free with M; no further structure was needed. There is a natural smooth map onto M:

$\text{p}:\text{T}\text{M}\to \to \text{M}:(\text{x},\text{u})\mapsto \text{x}$

The fiber of such a map over x is

${\text{p}}^{\leftarrow }\left\{\text{x}\right\}=\left\{\text{y}\in \text{T}\text{M}|\text{p}\left(\text{y}\right)=\text{x}\right\}$

which evidently coincides with T _x M and therefore looks like $\mathbb{R}$ ⁿ for each x, though there is no unique isomorphism T _xM≃ $\mathbb{R}$ ⁿ actually determined for each x since each chart gives a different one. The tangent bundle is an example of an important class of structures over manifolds, the vector bundles.

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Stable Homotopy and Iterated Loop Spaces

Gunnar Carlsson , R. James Milgram , in Handbook of Algebraic Topology, 1995

8.4 The combinatorial data which build Ω-spectra

These constructions also allow one to construct spectra from purely combinatorial data. To understand this, we recall the nerve construction, which associates to any category, C, a simplicial set, N.C, and hence a topological space. The k-simplices are composable k-tuples of arrows

$x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }{x}_2\stackrel{f_3}{\to }\cdots \stackrel{f_k}{\to }{x}_k$

in C if k > 0, and are simply objects in C if k = 0. The face maps are given by the following formulae.

$\{\begin{array}{ll}{d}_0\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }\cdots \stackrel{f_k}{\to }{x}_k\right)\hfill & =\left(x_1\stackrel{f_2}{\to }{x}_2\stackrel{f_3}{\to }\cdots \stackrel{f_k}{\to }{x}_k\right),\hfill \\ d_i\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }\cdots \stackrel{f_k}{\to }{x}_k\right)\hfill & =\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }\cdots x_{i-1}\stackrel{f_{i+1}○f_i}{\to }{x}_{i+1}\cdots \stackrel{f_k}{\to }{x}_k\right)\hfill \\ \hfill & \text{for}\text{1}\le i<k,\hfill \\ d_k\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }\cdots \stackrel{f_k}{\to }{x}_k\right)\hfill & =\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }{x}_2\cdots \stackrel{f_{k-1}}{\to }{x}_{k-1}\right),\hfill \\ S_i\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }\cdots \stackrel{f_k}{\to }{x}_k\right)\hfill & =\left(x_0\stackrel{f_1}{\to }{x}_1\stackrel{f_2}{\to }\cdots \to x_{i-1}\stackrel{f_i}{\to }{x}_i\stackrel{id}{\to }{x}_i\right)\hfill \\ \hfill & \stackrel{f_{i+1}}{\to }{x}_{i+1}\to \cdots \stackrel{f_k}{\to }{x}_k).\hfill \end{array}$

This is often a convenient way to construct spaces and maps, since it is clear that functors induce maps of simplicial sets. Indeed, any simplicial complex is homeomorphic to the nerve of a category, hence any CW complex has the homotopy type of the nerve of a suitable category. It is reasonable to ask what additional structure on the category allows one to construct a spectrum from N.C in the same way as the Q or ${\underset{\_}{T}}^B$ -algebra structures allowed one to construct spectra out of a space X. In order to describe this structure, we need a definition.

DEFINITION 8.4.1

A permutative category is a triple (C, ⊕, c), where C is a category, ⊕ : C × C → C is a functor, and c is a natural isomorphism of functors, from ⊕ to ⊕ ○ τ, where τ : C × C → C × C is the "reverse coordinates" map, subject to the following conditions.

(a)

⊕ is associative in the sense that

$\oplus ○\left(Id\times \oplus \right)=\oplus ○\left(\oplus \times Id\right).$

(b)

c(y, x) ○ c(x, y) = id _{(z, y)} for all (x, y) ε C × C.

(c)

The diagram

commutes.

(d)

c(A ⊕ *) = Id_A .

The nerve of a permutative category becomes a simplicial monoid. Further, its realization is a BE-algebra, where BE is the triple corresponding to the Barratt–Eccles $O$ -space |B|. To see this, one observes that $\underset{\_}{B}\left[|NC\right]|$ can itself be described as the realization of the nerve of a category, what one might call the free permutative category on C (see [34] or or [23]). One can now use the above described space level constructions to arrive at a connective spectrum Spt(C).

THEOREM 8.4.1

Spt defines a functor from the category of permutative categories to the category of connective spectra. Further, the zeroth space of Spt(C) has the homotopy type of the group completion of the monoid N ₀ C.

The last part of the statement is crucial for computations. It has as a corollary the well-known theorem of Barratt, Priddy and Quillen.

COROLLARY 8.4.1 ([28])

Let S _∞ denote the infinite symmetric group, i.e

$\underset{n}{\underrightarrow{\lim }}{S}_n,$

where S_n is included in S _n+1 in the evident way. Let $B{S}_{\infty }^{+}$ denote Quillen's plus construction on BSS _∞, which abelianizes the fundamental group without affecting homology. Then Q(S ⁰) ≅ BS ⁺ _∞ × Z. In particular, if Q(S ⁰)₀ denotes the component consisting of maps of degree 0, H _*(Q(S ⁰)₀;Z) ≅ H _*(BS _∞;Z).

PROOF

The Barratt–Eccles monoid valued construction on S ⁰, which is the nerve of a category with two objects * and p, and only identity morphisms, is isomorphic to ∐ _n≥0 BS_n , equipped with an associative multiplication, carrying BS_n × BS_m into BS _n+m . It is not hard to see that the group completion is homotopy equivalent to BS _∞ × Z. The result now follows from the above results.

We conclude with some examples.

(A)

The category of finite sets can be given the structure of a permutative category, with the sum operation corresponding to disjoint union. The resulting spectrum is the sphere spectrum.

(B)

Let G be a finite group, and consider the category of finite sets with G-action. As in (A) above, we obtain a permutative category, which corresponds to Segal's G-equivariant sphere spectrum. It is a bouquet of spectra parameterized by the conjugacy classes of subgroups K of G, where the summand corresponding to the conjugacy class of K is the suspension spectrum of the classifying space of the group N_G (K)/K.

(C)

Let A be any abelian group. View it as a category whose objects are the elements of A, and whose only morphisms are identity maps. The addition in A makes this category into a permutative category, in which c is actually an identity map for all pairs of objects in the category. The associated spectrum is the Eilenberg–MacLane spectrum K(A, 0).

(D)

Let R be any ring, and consider the category of all finitely generated projective R-modules. This can be given the structure of a permutative category, where the sum operation corresponds to direct sum of modules. The corresponding spectrum is Quillen's algebraic K-theory spectrum for the ring R.

Both authors were partially supported by grants from the N.S.F.

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Probabilities and Potential C

In North-Holland Mathematics Studies, 1988

A visitor from the future

56

We hope, in the subsequent chapters, to study canonical Markov processes and their translation operators. We shall indicate here an application of the procedure of no. 41, which is not in its proper place here, but rather belongs to volume D.

Let E be a separable metrizable space, and let Ω be a subset of the product space ${\text{E}}^{\mathcal{R}}+$ which is stable under translation (i.e. if a path ω belongs to Ω, so does the path ω(s+.) for all s – which we denote θ_Sω). We denote by X_t the coordinate map $\text{ω}\mapsto \text{ω}\left(\text{t}\right)$ on Ω, by $\mathcal{F}$ the σ-field generated by the X_t, and we make the following two hypotheses.

i)

The map $\left(\text{t},\text{ω}\right)\mapsto {\text{X}}_{\text{t}}\left(\text{ω}\right)$ from $\mathcal{R}$ ₊ × Ω to E is measurable (it is easily deduced that $\left(\text{t},\text{ω}\right)\mapsto {\text{θ}}_{\text{t}}\text{ω}$ is measurable from $\mathcal{R}$ ₊ × Ω to Ω).

ii)

The measurable space $\left(\text{Ω},\mathcal{F}\right)$ is cosouslinian.

For example, when E is cosouslinian and Ω is the space of right continuous paths with left limits from $\mathcal{R}$ ₊ to E, these hypotheses are satisfied (IV.19). In applications of the following result, one has in most cases G = E (which requires that E be Lusin) but the general case also merits treatment.

THEOREM. Let G be a Souslin measurable space, $\text{x}\mapsto {\text{P}}^{\text{x}}$ a submarkovian kernel from G to Ω. Then there exists a Lusin subspace Ω′ of Ω, carrying all the measures P^x (x ∈ G) and stable under translation.

Proof. We shall construct an increasing sequence (Ω_n) of subsets of Ω, with the following properties:

–

Ω₀ carries all the measures P^x

–

for even n, Ω_n is Lusin

–

for odd n, Ω_n is Souslin, and stable under translation; Ω′ will be their union.

We begin with Ω₀. Thanks to III.19-20, we identify Ω with a coanalytic subset F of a compact metrizable space $\overline{\text{F}}$ , $\mathcal{F}$ with the σ-field induced by $\text{B}\left(\overline{\text{F}}\right)$ . The set H = {P^x,x ∈ G) is then identified with an analytic set of measures on $\overline{\text{F}}$ carried by Ω. By the argument of 41 (with the same notation) there exists a Borel set B in $\overline{\text{F}}$ , containing $\overline{\text{F}}\backslash \text{F}$ and such that P^x(B) = 0 for all × ∈ G, and we set Ω₀ = B^C.

We now proceed by induction:

1)

For even n, Ω_n+1 will be the image of the map $\left(\text{t},\text{ω}\right)\mapsto 0_{\text{t}}\text{ω}$ from $\mathcal{R}$ ₊ × Ω_n to Ω: it is Souslin, stable under translation, and contains Ω_n.

2)

For odd n, Ω_n and Ω^C are two disjoint analytic subsets that may be separated by a Borel set Ω_n+1, so that Ω_n ⊂ Ω_n+1 ⊂ Ω.

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Prevalence

Brian R. Hunt , Vadim Yu. Kaloshin , in Handbook of Dynamical Systems, 2010

2.3.2 Projections and embeddings

In this section we discuss results of the following type. Given a space of functions from a set $S$ of a Euclidean space, what properties of $S$ or measure $\mu$ on $S$ are preserved in its image under a generic or prevalent function?

If $S$ is a $C^r$ manifold of dimension $d$ for $r\ge 1$ , the Whitney Embedding Theorem [185] says that $S$ can be embedded in ${\mathbb{R}}^{2d+1}$ . Whitney's proof is perturbative and implies that a generic $C^r$ function from $S$ to ${\mathbb{R}}^n$ is an embedding (that is, a diffeomorphism between $S$ and its image) for $n>2d$ . In this case all diffeomorphism-invariant properties of $S$ are preserved in its image. The proof of Mañé's result [106] for linear projections (see Example 8 in Section 1.2) can be modified to show that if $S$ is a compact subset of a Banach space with box-counting dimension $d$ , then if $n>2d+1$ , a generic $C^r$ function in ${\mathbb{R}}^n$ is one-to-one. Sauer, Yorke, and Casdagli [154] proved Whitney's result for a prevalent $C^r$ function, and showed that if $S$ is a compact subset of a Euclidean space with box-counting dimension $d$ , a prevalent $C^r$ function (defined at least on a neighbourhood of $S$ ) in ${\mathbb{R}}^n$ is an embedding on $S$ , again provided that $n>2d$ . (Here and below, by 'embedding' on a set $S$ that is not necessarily a manifold, we mean that the function is one-to-one and is an embedding on every compact manifold-with-boundary in $S$ .)

Takens [174] proved a dynamical Embedding Theorem, which states that for a $C^2$ flow ${\left\{{\Phi }_t\right\}}_{t\in \mathbb{R}}$ on a $d$ -dimensional $C^2$ manifold $S$ subject to certain nondegeneracy conditions, a generic $C^2$ function $h$ from $S$ to $\mathbb{R}$ generates an embedding into ${\mathbb{R}}^{2d+1}$ when composed with the flow as follows:

$x\mapsto (h\left(x\right),h\left({\Phi }_{-1}\left(x\right)\right),h\left({\Phi }_{-2}\left(x\right)\right),\dots ,h\left({\Phi }_{-2d}\left(x\right)\right))\text{.}$

Such a map is called a 'delay-coordinate embedding'. Sauer, Yorke, and Casdagli [154] proved a version of this result for prevalent $h$ , and showed that if $S$ is not a manifold but instead a compact invariant subset of ${\mathbb{R}}^m$ with box-counting dimension $d$ , then the delay-coordinate map into ${\mathbb{R}}^n$ defined by a $C^2$ function $h:{\mathbb{R}}^m\to \mathbb{R}$ is prevalently an embedding if $n>2d$ . Their non-degeneracy condition, somewhat weaker than Takens', is that the flow has finitely many fixed points, finitely many periodic orbits with integer period at most $n$ , and no periodic orbits with period 1 or 2.

In the analysis of experimental data believed to arise from a chaotic attractor, a common technique is to try to embed the attractor into ${\mathbb{R}}^n$ , either by making $n$ simultaneous measurements of the state of the system at a series of times, or by using the delay-coordinate technique with a single measurement made at each time. In such cases the attractor dimension is not known a priori, so one cannot know whether the embedding results above actually apply. Ott and Yorke [136] formulated versions of the prevalent results above in which the hypotheses can be checked on the image of $S$ rather than on $S$ . For example, with an appropriate definition of 'tangent dimension', they showed that a prevalent $C^1$ function from a compact subset $S$ of a Euclidean space into ${\mathbb{R}}^n$ has the property that if the tangent dimension of the image of $S$ is less than $n/2$ , then the function is an embedding on $S$ .

In cases where $S$ is mapped to ${\mathbb{R}}^n$ where $n$ is not sufficiently large to guarantee that the mapping is generically or prevalently one-to-one, one may still expect certain features of $S$ to be preserved in its image, for example its dimension. A classical result [110,91,115] is that if $S$ is a compact subset of ${\mathbb{R}}^m$ with Hausdorff dimension at most $n$ , then almost every orthogonal projection onto an $n$ -dimensional subspace preserves the Hausdorff dimension of $S$ . Here 'almost every' is with respect to the measure on the Grassmanian manifold of $n$ -dimensional subspaces of ${\mathbb{R}}^m$ induced by Haar measure on the orthogonal group $O\left(m\right)$ . We can restate this result in terms of Lebesgue measure by considering linear transformations from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ . Since full-rank matrices have full Lebesgue measure in the space of $n\times m$ matrices, and every full-rank linear transformation from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ can be expressed in a unique way as a composition of an orthogonal projection onto an $n$ -dimensional subspace and a linear isomorphism from that subspace to ${\mathbb{R}}^n$ , it follows that Lebesgue almost every linear transformation from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ preserves the Hausdorff dimension of $S$ (again provided that this dimension is at most $n$ ).

Using prevalence, one can extend these results to spaces of nonlinear functions and/or to the case that $S$ lies in an infinite-dimensional space. Sauer and Yorke [153] proved that for compact $S\subset {\mathbb{R}}^m$ with Hausdorff dimension at most $n$ , a prevalent $C^1$ function from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ preserves the Hausdorff dimension of $S$ , and that the same is true for the correlation dimension of a compactly supported measure on ${\mathbb{R}}^m$ . (They also gave an example showing that no such result is possible for a box-counting dimension.) They also showed that for a compactly supported invariant measure $\mu$ of a diffeomorphism $F:{\mathbb{R}}^m\to {\mathbb{R}}^m$ , the correlation dimension of $\mu$ is preserved by the delay-coordinate map

$x\mapsto (h\left(x\right),h\left(F^{-1}\left(x\right)\right),h\left(F^{-2}\left(x\right)\right),\dots ,h\left(F^{-(n-1)}\left(x\right)\right))\text{.}$

for a prevalent $C^1$ function $h:{\mathbb{R}}^m\to \mathbb{R}$ , provided that $n$ is at least the correlation dimension of $\mu$ and that the set of periodic points of $F$ with period at most $n$ has Lebesgue measure zero.

In the case that $S$ is a compact subset of a Banach space $B$ , Hunt and Kaloshin [73] proved that if $n$ is greater than twice the box-counting dimension of $S$ and $V$ is a space of functions from $B$ to ${\mathbb{R}}^n$ that contains the bounded linear functions and is contained in the locally Lipschitz functions, then a prevalent function in $V$ is one-to-one on $S$ . Thus, the conditions under which a prevalent function is one-to-one on $S$ are the same as in the case where $S$ lies in a Euclidean space. On the other hand, for each real $d$ they gave, in the same paper, an example of a compact set $S_0$ with Hausdorff dimension $d$ in a Hilbert space, such that the image of $S_0$ under every bounded linear function into ${\mathbb{R}}^n$ has Hausdorff dimension less than $d$ . With an additional assumption on the compact subset $S$ of Banach space $B$ , Ott, Hunt, and Kaloshin [134] proved that the Hausdorff dimension of $S$ is preserved by a prevalent function from any of the function spaces $V$ described above.

Kahane [79] (see also [56]) proved some results about images of sets and measures under prevalent continuous (rather than smooth) functions that differ strikingly from their generic counterparts. Specifically, he showed that given a Cantor set (a homeomorphic image of the canonical middle-third Cantor set) $S$ in $\mathbb{R}$ , the image of $S$ under a prevalent continuous function $f$ from $\mathbb{R}$ to itself is the closure of its interior, i.e. $f\left(S\right)=\overline{\text{I}ntf\left(S\right)}$ , despite the fact that $S$ itself has no interior. By contrast, Kaufmann [90] showed that the image of $S$ under a generic continuous function is a Cantor set with Lebesgue measure zero. Kahane also proved that given a compactly supported non-atomic measure $\mu$ , the image measure $\mu \circ f^{-1}$ under a prevalent continuous function $f$ is absolutely continuous (with respect to Lebesgue measure) with $C^{\infty }$ density. By contrast, the image of $\mu$ under a generic continuous function is singular with respect to Lebesgue measure.

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Probabilities and Potential C

In North-Holland Mathematics Studies, 1988

A Method of Compactification

47.

We consider a measure space (E,ε) which we shall suppose Lusin or Souslin (III.16). Let Γbe a convex, ∧-stable cone of bounded, Borel functions on E, containing the positive constants and separating points, separable for uniform convergence (we shall denote by (γ_n) a dense sequence in Γ). We define as in no. 28 the sweeping relation ┤_Γ, the gambling house J …

It is possible to establish a certain number of results on the sweeping relative to Γ, by a reduction to the preceding theory using a compactification procedure, which constitutes a very effective method in potential theory (a method more interesting than what we shall deduce from it here).

a)

Let h be the measurable, injective map × ↦ (γ_n(x))_n∈N from E to ${\text{R}}^{\mathcal{N}}$ ; by III.21 the image h(E) is Borel in ${\mathcal{R}}^{\mathcal{N}}$ (Souslin if E is Soulsin) and h establishes an isomorphism between ε and the Borel σ-field of h(E). So there is no harm in identifying E with its image.

Each function γ_n being bounded, the closure $\overline{\text{E}}$ of E is compact in ${\mathcal{R}}^{\mathcal{N}}$ . As γ_n is identified with the nth coordinate map, it admits a continuous extension to $\overline{\text{E}}$ (unique since E is dense in $\overline{\text{E}}$ ). This property extends by uniform convergence to all f ∈ Γ: we will denote by $\overline{\text{f}}$ the corresponding extension, and by $\overline{\text{Γ}}$ the set of such extensions. It is clear that $\overline{\text{Γ}}$ separates points of $\overline{\text{E}}$ . $\overline{\text{Γ}}$ is a ∧-stable convex cone, which contains the positive constants. By the Stone-Weierstrass theorem $\overline{\text{Γ}}\overline{\text{Γ}}$ is dense in $\text{e}\left(\overline{\text{E}}\right)$ .

Until this last remark we have not used the fact that Γ is a convex cone, nor the stability under ∧.

b)

The gambling house $\overline{\text{J}}$ associated with $\overline{\text{Γ}}$ is compact; the gambling house J is identified with the set of pairs $\left(\text{x,μ}\right)\in \overline{\text{J}}$ such that × ∈ E and μ is carried by E. It is not hard to show (III.60) that J is analytic. We can thus apply the results of §1 to J and $\overline{\text{J}}$ .

Let λ and μ be two positive measures on E, identified to measures on $\overline{\text{E}}$ carried by E). It is clear that the relations $\text{λ}{⊣}_{\overline{\text{Γ}}}\text{μ}$ and $\text{λ}{⊣}_{\text{Γ}}\text{μ}$ are equivalent, and similarly $\text{λ}{⊣}_{{\overline{\text{Γ}}}_{+}}\text{μ}$ and $\text{λ}{⊣}_{{\text{Γ}}_{+}}\text{μ}$ . So, applying 39, we see that $\text{λ}{⊣}_{\text{J}}\text{μ}$ is equivalent to $\text{λ}{⊣}_{\overline{\text{J}}}\text{μ}$ . A little more generally, the restriction to E of each $\overline{\text{J}}$ -sweeping of λ is a J-sweeping of λ.

If h is a $\overline{\text{J}}$ -supermedian function on $\overline{\text{E}}$ , its restriction to E is J-supermedian. Conversely every analytic J-supermedian function f on E is the restriction to E of an analytic $\overline{\text{J}}$ -supermedian function on $\overline{\text{E}}$ : let f⁰ be the extension of f to $\overline{\text{E}}$ which is zero on E^c; as {f⁰ > t} = {f > t} is analytic in E, hence in $\overline{\text{E}}$ , f⁰ is analytic and so is its reduction $\overline{\text{R}}\left({\text{f}}^0\right)$ , and it suffices to show $\overline{\text{R}}\left({\text{f}}^0\right)=\text{f}$ (or simply ≤ f) on E. But, at each point × ∈ E, the left hand side equals ${\text{sup}}_{\text{μ}\in {\overline{\text{J}}}_{\text{x}}}\text{μ}\left({\text{f}}^0\right);$ we can replace μ(f⁰) by η(f), where η is the restriction of μ to E, and as ε_x┤_J η and f is J-supermedian this quantity is indeed majorized by f(x). More generally, if f is analytic we have $\text{R}\text{f}=\overline{\text{R}}\left({\text{f}}^0\right)$ on E.

The compactification method can be used to extend theorem 39: if $\text{λ}{⊣}_{{\text{Γ}}_{\text{+}}}\text{μ}$ there exists ν ≥ μ such that λ┤_Γ ν (it is enough to apply 39 on $\overline{\text{E}}$ and to take a restriction to E).

Another consequence is the extension of Strassen's theorem 40: if λ and μ are carried by E, and λ┤_Γ μ, there exists a kernel $\overline{\text{p}}$ , permitted in $\overline{\text{J}}$ , such that $\text{μ}=\text{λ}\overline{\text{P}}$ . But since $\text{λ}\overline{\text{P}}=\text{μ}$ is carried by E, the set of those × such that ${\text{ε}}_{\text{x}}\overline{\text{P}}$ is not carried by E is λ-negligible; we enclose it in a λ-negligible Borel set B, and we set

${\text{ε}}_{\text{x}}\text{P}={\text{ε}}_{\text{x}}\overline{\text{P}}\text{f}\text{o}\text{r}\text{x}\text{ε}\text{E}\backslash \text{B}{\text{ε}}_{\text{x}}\text{P}={\text{ε}}_{\text{x}}\text{f}\text{o}\text{r}\text{x}\in \text{E}\cap \text{B}$

which gives a permitted kernel P in J such that μ ▭ λP.

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Geodesy

Petr Vaníček , in Encyclopedia of Physical Science and Technology (Third Edition), 2003

II.C Relative Positioning

Relative positioning, meaning positioning of a point with respect to an existing point or points, is the preferred mode of positioning in geodesy. If there is intervisibility between the points, terrestrial techniques can be used. For satellite relative positioning, the intervisibility is not a requirement, as long as the selected satellites are visible from the two points in question. The accuracy of such relative positions is usually significantly higher than the accuracy of single point positions.

The classical terrestrial techniques for 2D relative positioning make use of angular (horizontal) and distance measurements, which always involve two or three points. These techniques are thus differential in nature. The computations of the relative 2D positions are carried out either on the horizontal datum (reference ellipsoid), in terms of latitude difference Δϕ and longitude difference Δλ, or on a map, in terms of Cartesian map coordinate differences Δx and Δy. In either case, the observed angles, azimuths, and distances have to be first transformed (reduced) from the earth's surface, where they are acquired, to the reference ellipsoid, where they are either used in the computations or transformed further onto the selected mapping plane. We shall not explain these reductions here; rather, we would advise the interested reader to consult one of the classical geodetic textbooks (e.g.,Zakatov, 1953;Bomford, 1971).

To determine the relative position of one point with respect to another on the reference ellipsoid is not a simple proposition, since the computations have to be carried out on a curved surface and Euclidean geometry no longer applies. Links between points can no longer be straight lines in the Euclidean sense; they have to be defined as geodesics (the shortest possible lines) on the reference ellipsoid. Consequently, closed form mathematical expressions for the computations do not exist, and use has to be made of various series approximations. Many such approximations had been worked out, which are valid for short, medium, or long geodesics. For 200 years, coordinate computations on the ellipsoid were considered to be the backbone of (classical) geodesy, a litmus test for aspiring geodesists. Once again, we shall have to desist from explaining the involved concepts here as there is no room for them in this small article. Interested readers are referred once more to the textbooks cited above.

Sometimes, preference is given to carrying out the relative position computations on the mapping plane, rather than on the reference ellipsoid. To this end, a suitable cartographic mapping is first selected, normally this would be the conformal mapping used for the national/state geodetic work. This selection carries with it the appropriate mathematical mapping formulae and distortions associated with the selected mapping (Lee, 1976). The observed angles ω, azimuths α, and distances S (that had been first reduced to the reference ellipsoid) are then reduced further (distorted) onto the selected mapping plane where (2D) Euclidean geometry can be applied. This is shown schematically inFig. 4. Once these reductions have been carried out, the computation of the (relative) position of the unknown point B with respect to point A already known on the mapping plane is then rather trivial:

FIGURE 4. Mapping of ellipsoid onto a mapping plane.

(7) $\begin{array}{cc}\hfill {\text{x}}_{\text{B}}={\text{x}}_{\text{A}}+\mathrm{\Delta }{\text{x}}_{\text{AB}},& \hfill {\text{y}}_{\text{B}}={\text{y}}_{\text{A}}+\mathrm{\Delta }{\text{y}}_{\text{AB}}.\hfill \end{array}$

Relative vertical positioning is based on somewhat more transparent concepts. The process used for determining the height difference between two points is called geodetic levelling (Bomford, 1971). Once the levelled height difference is obtained from field observations, one has to add to it a small correction based on gravity values along the way to convert it to either the orthometric, the dynamic, or the normal height difference. Geodetic levelling is probably the most accurate geodetic relative positioning technique. To determine the geodetic height difference between two points, all we have to do is to measure the vertical angle and the distance between the points. Some care has to be taken that the vertical angle is reckoned from a plane perpendicular to the ellipsoidal normal at the point of measurement.

Modern extraterrestrial (satellite and radio astronomical) techniques are inherently three dimensional. Simultaneous observations at two points yield 3D coordinate differences that can be added directly to the coordinates of the known point A on the earth's surface to get the sought coordinates of the unknown point B (on the earth's surface). Denoting the triplet of Cartesian coordinates (x, y, z) in any coordinate system by r and the triplet of coordinate differences (Δx, Δy, Δz) by Δr, the 3D position of point B is given simply by

(8) ${\mathbf{r}}_{\text{B}}={\mathbf{r}}_{\text{B}}+\mathrm{\Delta }{\mathbf{r}}_{\text{AB}},$

where Δr _AB comes from the observations.

We shall discuss inSection V.B how the "base vector" Δr _AB is derived from satellite observations. Let us just mention here that Δr _AB can be obtained also by other techniques, such as radio astronomy, inertial positioning, or simply from terrestrial observations of horizontal and vertical angles and distances. Let us show here the principle of the interesting radio astronomic technique for the determination of the base vector, known in geodesy as Very Long Baseline Interferometry (VLBI).Figure 5 shows schematically the pair of radio telescopes (steerable antennas, A and B) following the same quasar whose celestial position is known (meaning that e _s is known). The time delay τ can be measured very accurately and the base vector Δr _AB can be evaluated from the following equation:

FIGURE 5. Radioastronomical interferometry.

(9) $\tau ={\text{c}}^{-1}{\mathbf{e}}_{\text{s}}\mathrm{\Delta }{\mathbf{r}}_{\text{AB}},$

where c is the speed of light. At least three such equations are needed for three different quasars to solve for Δr _AB.

Normally, thousands of such equations are available from dedicated observational campaigns. The most important contribution of VLBI to geodesy (and astronomy) is that it works with directions (to quasars) which can be considered as the best approximations of directions in an inertial space.

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