ONT Re: Manifolds Of Sensuous Impressions (MOSI's)
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| Now, our supposed little feeling gives a 'what';
| and if other feelings should succeed which remember the first,
| its 'what' may stand as subject or predicate of some piece of knowledge-about,
| of some judgment, perceiving relations between it and other 'whats' which the other
| feelings may know. The hitherto dumb 'q' will then receive a name and be no longer speechless.
|
| James, "Func of Cog", page 14.
|
| William James, "The Function Of Cognition",
| Read before the Aristotelian Society, 1 Dec 1884.
| First published in 'Mind', 10 (1885). Reprinted in
|'The Meaning Of Truth, A Sequel To "Pragmatism"',
| Longmans, Green, & Company, London, UK, 1909.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| As with immersions and submersions, we have a characterization
| of transversal maps in terms of tangent spaces.
|
| Proposition 2.4. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces.
|
| Let f : X -> Y be a C^p-morphism, and W a submanifold of Y.
|
| The map f is transversal over W
|
| if and only if
|
| for each x in X such that f(x) lies in W,
|
| the composite map
|
| T_x(f)
| T_x(X) --------> T_w(Y) ------> T_w(Y)/T_w(W)
|
| with w = f(x) is surjective and its kernel splits.
|
| Proof. If f is transversal over W, then for each point x in X such that f(x) lies in W,
| we choose charts as in the definition, and reduce the question to one of maps
| of open subsets of Banach spaces. In that case, the conclusion concerning the
| tangent spaces follows at once from the assumed direct product decompositions.
| Conversely, assume our condition on the tangent map. The question being local,
| we can assume that Y = V_1 x V_2 is a product of open sets in Banach spaces
| such that W = V_1 x 0, and we can also assume that X = U is open in some
| Banach space, x = 0. Then we let g : U -> V_2 be the map pi o f, where
| pi is the projection, and note that our assumption means that g'(0) is
| surjective and its kernel splits. Furthermore, g^-1(0) = f^-1(W).
| We can then use Corollary 5.7 of the inverse mapping theorem
| to conclude the proof.
|
| Lang, DARM, pages 27-28.
¤~~~~~~~~~¤~~~~~~~~~¤~SYLLABUS~¤~~~~~~~~~¤~~~~~~~~~¤
| 2. Manifolds
|
| 2.1. Atlases, Charts, Morphisms
|
| Let X be a set. An "atlas of class C^p (p >= 0)" on X is a collection
| of pairs (U_i, q_i) (i ranging in some indexing set), satisfying the
| following conditions:
|
| AT 1. Each U_i is a subset of X and the U_i cover X.
|
| AT 2. Each q_i is a bijection of U_i onto an open subset q_i U_i
| of some Banach space E_i and for any i, j, [it holds that]
| q_i (U_i |^| U_j) is open in E_i.
|
| AT 3. The map
|
| q_j o q_i^-1 : q_i (U_i |^| U_j) --> q_j (U_i |^| U_j)
|
| is a C^p-isomorphism for each pair of indices i, j.
|
| Lang, DARM, page 20.
|
| Each pair (U_i, q_i) will be called a "chart" of the atlas.
| If a point x of X lies in U_i, then we say that (U_i, q_i)
| is a "chart at" x.
|
| Suppose that we are given an open subset U of X and a topological isomorphism
| q : U -> U' onto an open subset of some Banach space E. We shall say that
| (U, q) is "compatible" with the atlas {(U_i, q_i)} if each map q_i q^-1
| (defined on a suitable intersection as in AT 3) is a C^p-isomorphism.
|
| Two atlases are said to be "compatible" if each chart of one is compatible with
| the other atlas. One verifies immediately that the relation of compatibility
| between atlases is an equivalence relation. An equivalence class of atlases
| of class C^p on X is said to define a structure of "C^p-manifold" on X.
|
| If all the vector spaces E_i in some atlas are toplinearly isomorphic,
| then we can always find an equivalent atlas for which they are all equal,
| say to the vector space E. We then say that X is an "E-manifold" or that
| X is "modeled" on E.
|
| If E = R^n for some fixed n, then we say that the manifold is "n-dimensional".
| In this case, a chart
|
| q : U -> R^n
|
| is given by n coordinate functions q_1, ..., q_n. If 'P' denotes a point of U,
| these functions are often written
|
| x_1(P), ..., x_n(P),
|
| or simply x_1, ..., x_n. They are called "local coordinates" on the manifold.
|
| If the integer p (which may also be infinity) is fixed throughout a discussion,
| we also say that X is a manifold.
|
| Lang, DARM, page 21.
|
| Let X be manifold, and U an open subset of X. Then it is possible,
| in the obvious way, to induce a manifold structure on U, by taking
| as charts the intersections
|
| (U_i |^| U, q_i | (U_i |^| U)).
|
| [Notation. "f | S" indicates the function f as restricted to the set S.]
|
| If X is a topological space, covered by open subsets V_j, and if we are
| given on each V_j a manifold structure such that for each pair j, j' the
| induced structure on V_j |^| V_j' coincides, then it is clear that we can
| give to X a unique manifold structure inducing the given ones on each V_j.
|
| If X, Y are two manifolds, then one can give the
| product X x Y a manifold structure in the obvious way.
| If {(U_i, q_i)} and {(V_j, r_j)} are atlases for X, Y
| respectively, then
|
| {(U_i x V_j, q_i x r_j)}
|
| is an atlas for the product, and the product of compatible
| atlases gives rise to compatible atlases, so that we do get
| a well-defined product structure.
|
| Let X, Y be two manifolds. Let f : X -> Y be a map.
| We shall say that f is a "C^p-morphism" if, given x in X,
| there exists a chart (U, q) at x and a chart (V, r) at f(x)
| such that f(U) c V, and the map
|
| r o f o q^-1 : qU -> rV
|
| is a C^p-morphism in the sense of Chapter 1, Section 3.
| One sees then immediately that this same condition holds
| for any choice of charts (U, q) at x and (V, r) at f(x)
| such that F(U) c V.
|
| It is clear that the composite of two C^p-morphisms is itself
| a C^p-morphism (because it is true for open subsets of vector
| spaces). The C^p-manifolds and C^p-morphisms form a category.
| The notion of isomorphism is therefore defined ...
|
| If f : X -> Y is a morphism, and (U, q) is a chart
| at a point x in X, while (V, r) is a chart at f(x),
| then we shall also denote by
|
| f_V,U : qU -> rV
|
| the map rfq^-1 [that is, r o f o q^-1].
|
| Lang, DARM, page 22.
|
| It is also convenient to have a local terminology.
| Let U be an open set (of a manifold or a Banach space)
| containing a point x_0. By a "local isomorphism" at x_0
| we mean an isomorphism
|
| f : U_1 -> V
|
| from some open set U_1 containing x_0 (and contained in U)
| to an open set V (in some manifold or some Banach space).
| Thus a local isomorphism is essentially a change of chart,
| locally near a given point.
|
| 2.2. Submanifolds, Immersions, Submersions
|
| Let X be a topological space, and Y a subset of X.
| We say that Y is "locally closed" in X if every point
| y in Y has an open neighborhood U in X such that Y |^| U
| is closed in U. One verifies easily that a locally closed
| subset is the intersection of an open set and a closed set.
| For instance, any open subset of X is locally closed, and
| any open interval is locally closed in the plane.
|
| Let X be a manifold (of class C^p with p >= 0). Let Y be a subset of X
| and assume that for each point y in Y there exists a chart (V, r) at y
| such that r gives an isomorphism of V with a product V_1 x V_2 where
| V_1 is open in some space E_1 and V_2 is open in some space E_2,
| and such that
|
| r(Y |^| V) = V_1 x a_2
|
| for some point a_2 in V_2 (which we could take to be 0). Then it is clear
| that Y is locally closed in X. Furthermore, the map r induces a bijection
|
| r_1 : Y |^| V -> V_1.
|
| The collection of pairs (Y |^| V, r_1) obtained in the above manner constitues
| an atlas for Y, of class C^p. The verification of this assertion, whose formal
| details we leave to the reader, depends on the following obvious fact.
|
| Lang, DARM, page 23.
|
| Lemma 2.1. Let U_1, U_2, V_1, V_2 be open subsets of Banach spaces,
|
| and g : U_1 x U_2 -> V_1 x V_2 a C^p-morphism.
|
| Let a_2 be in U_2 and b_2 be in V_2
|
| and assume that g maps U_1 x a_2 into V_1 x b_2.
|
| Then the induced map
|
| g_1 : U_1 -> V_1
|
| is also a morphism.
|
| Indeed, it is obtained as a composite map
|
| U_1 -> U_1 x U_2 -> V_1 x V_2 -> V_1,
|
| the first map being an inclusion and the third a projection.
|
| We have therefore defined a C^p-structure on Y which will be called
| a "submanifold" of X. This structure satisfies a universal mapping
| property, which characterizes it, namely:
|
| | Given any map f : Z -> X from a manifold Z into X such that
| | f(Z) is contained in Y. Let f_Y : Z -> Y be the induced map.
| | Then f is a morphism if and only if f_Y is a morphism.
|
| The proof of this assertion depends on Lemma 2.1, and is trivial.
|
| Finally, we note that the inclusion of Y into X is a morphism.
|
| If Y is also a closed subspace of X, then
| we say that it is a "closed submanifold".
|
| Suppose that X is finite dimensional of dimension n, and that Y
| is a submanifold of dimension m. Then from the definition we see that
| the local product structure in the neighborhood of a point of Y can be
| expressed in terms of local coordinates as follows. Each point P of Y
| has an open neighborhood U in X with local coordinates (x_1, ..., x_n)
| such that the points of Y in U are precisely those whose last n - m
| coordinates are 0, that is, those points having coordinates of type
|
| (x_1, ..., x_m, 0, ..., 0).
|
| Let f : Z -> X be a morphism, and let z be in Z. We shall say that f is
| an "immersion" at z if there exists an open neighborhood Z_1 of z in Z
| such that the restriction of f to Z_1 induces an isomorphism of Z_1
| onto a submanifold of X. We say that f is an "immersion" if it is
| an immersion at every point.
|
| Lang, DARM, page 24.
|
| Note that there exist injective immersions
| which are not isomorphisms onto submanifolds,
| as given by the following example:
| ________
| / \
| / \
| | |
| | |
| \ V
| \___________________________________________
|
| (The arrow means that the line approaches itself without touching.)
| An immersion which does give an isomorphism onto a submanifold is
| called an "embedding", and it is called a "closed embedding" if
| this submanifold is closed.
|
| A morphism f : X -> Y will be called a "submersion" at a point x in X
| if there exists a chart (U, q) at x and a chart (V, r) at f(x) such that
| q gives an isomorphism of U on a product U_1 x U_2 (U_1 and U_2 open in
| some Banach spaces), and such that the map
|
| rfq^-1 = f_V,U : U_1 x U_2 -> V
|
| is a projection. One sees then that the image of a submersion is
| an open subset (a submersion is in fact an open mapping). We say
| that f is a "submersion" if it is a submersion at every point.
|
| For manifolds modeled on Banach spaces, we have the usual criterion
| for immersions and submersions in terms of the derivative.
|
| Proposition 2.2. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces.
|
| Let f : X -> Y be a C^p-morphism. Let x be in X. Then:
|
| 1. f is an immersion at x if and only if
|
| there exists a chart (U, q) at x and (V, r) at f(x)
|
| such that f'_V,U(qx) is injective and splits.
|
| 2. f is a submersion at x if and only if
|
| there exists a chart (U, q) at x and (V, r) at f(x)
|
| such that f'_V,U(qx) is surjective and its kernel splits.
|
| Proof. This is an immediate consequence
| of Corollaries 5.4 and 5.6 of
| the inverse mapping theorem.
|
| The conditions expressed in [Propositions 2.2.1 and 2.2.2] depend only on the
| derivative [f'], and if they hold for one choice of charts (U, q) and (V, r),
| respectively, then they hold for every choice of such charts. It is therefore
| convenient to introduce a terminology in order to deal with such properties.
|
| Let X be a manifold of class C^p (p >= 1).
| Let x be a point of X. We consider triples (U, q, v)
| where (U, q) is a chart at x and v is an element of the
| vector space in which qU lies. We say that two such triples
| (U, q, v) and (V, r, w) are "equivalent" if the derivative of
| rq^-1 at qx maps v on w. The formula reads:
|
| (rq^-1)'(qx)v = w
|
| (obviously an equivalence relation by the chain rule).
|
| Lang, DARM, page 25.
|
| An equivalence class of such triples is called a "tangent vector" of X at x.
| The set of such tangent vectors is called the "tangent space" of X at x and
| is denoted by 'T_x(X)'. Each chart (U, q) determines a bijection of T_x(X)
| on a Banach space, namely the equivalence class of (U, q, v) corresponds to
| the vector v. By means of such a bijection it is possible to transport to
| T_x(X) the structure of topological vector space given by the chart, and
| it is immediate that this structure is independent of the chart selected.
|
| If U, V are open in Banach spaces, then to every morphism of class C^p (p >= 1)
| we can associate its derivative Df(x). If now f : X -> Y is a morphism of
| one manifold into another, and x a point of X, then by means of charts
| we can interpret the derivative of f on each chart at x as a mapping
|
| df(x) = T_x f : T_x(X) -> T_f(x)(Y).
|
| Indeed, this map T_x f is the unique linear map having the following property.
| If (U, q) is a chart at x and (V, r) is a chart at f(x) such that f(U) c V
| and ^v^ is a tangent vector at x represented by v in the chart (U, q), then
|
| T_x f(^v^)
|
| is the tangent vector at f(x) represented by Df_V,U(x)v.
| The representation of T_x f on the spaces of charts can
| be given in the form of a diagram
|
| T_x(X) o-------->o E
| | |
| T_x f | | f'_V,U(x)
| v v
| T_f(x)(Y) o-------->o F
|
| The map T_x f is obviously continuous and linear for the structure of
| topological vector space which we have placed on T_x(X) and T_f(x)(Y).
|
| As a matter of notation, we shall sometimes write f_*,x instead of T_x f.
|
| The operation T satisfies an obvious functorial property,
| namely, if f : X -> Y and g : Y -> Z are morphisms, then
|
| T_x(g o f) = T_f(x)(g) o T_x(f).
|
| T_x(id) = id.
|
| We may reformulate Proposition 2.2:
|
| Proposition 2.3. Let X, Y be manifolds of class C^p (p >= 1) modeled on Banach spaces.
|
| Let f : X -> Y be a C^p-morphism. Let x be in X. Then:
|
| 1. f is an immersion at x if and only if
|
| the map T_x f is injective and splits.
|
| 2. f is a submersion at x if and only if
|
| the map T_x f is surjective and its kernel splits.
|
| Note. If X, Y are finite dimensional, then the condition that T_x f splits
| is superfluous. Every subspace of a finite dimensional vector space splits.
|
| Lang, DARM, page 26.
|
| If W is a submanifold of a manifold Y of class C^p (p >= 1), then the inclusion
|
| i : W -> Y
|
| induces a map
|
| T_w i : T_w(W) -> T_w(Y)
|
| which is in fact an injection. From the definition of a submanifold, one sees
| immediately that the image of T_w i splits. It will be convenient to identify
| T_w(W) in T_w(Y) if no confusion can result.
|
| A morphism f : X -> Y will be said to be "transversal" over the submanifold W of Y
| if the following condition is satisfied.
|
| Let x in X be such that f(x) is in W. Let (V, r) be a chart at f(x) such that
| r : V -> V_1 x V_2 is an isomorphism on a product, with
|
| r(f(x)) = (0, 0) and r(W |^| V) = V_1 x 0.
|
| Then there exists an open neighborhood U of x such that the composite map
|
| f r pr
| U -----> V -----> V_1 x V_2 ------> V_2
|
| is a submersion.
|
| In particular, if f is transversal over W, then f^-1(W) is a submanifold of X,
| because the inverse image of 0 by our local composite map
|
| pr o r o f
|
| is equal to the inverse image of W |^| V by r.
|
| Lang, DARM, page 27.
|
| Serge Lang,
|'Differential & Riemannian Manifolds',
| Springer-Verlag, New York, NY, 1995.
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤
| o---------------------------------------o o-------------------o
| | X | | E_i |
| | | | |
| | | | o |
| | | | / \ |
| | o | | / \ |
| | / \ | | / \ |
| | / \ | | / \ |
| | / \ q_i | | / q_i U_i \ |
| | / o---------------------->| o o o |
| | / \ | | \ / \ / |
| | / \ | | \ / \ / |
| | / U_i \ | | o Eij o |
| | / \ | | \ / |
| | / \ | | \ / |
| | o o o | | o |
| | \ / \ / | | |
| | \ / \ / | | |
| | \ / \ / | o---------|---------o
| | \ / \ / | |
| | o Uij o | q_j o q_i^-1
| | / \ / \ | |
| | / \ / \ | o---------v---------o
| | / \ / \ | | E_j |
| | / \ / \ | | |
| | o o o | | o |
| | \ / | | / \ |
| | \ / | | / \ |
| | \ U_j / | | o Eji o |
| | \ / | | / \ / \ |
| | \ / | | / \ / \ |
| | \ o---------------------->| o o o |
| | \ / q_j | | \ q_j U_j / |
| | \ / | | \ / |
| | \ / | | \ / |
| | o | | \ / |
| | | | \ / |
| | | | o |
| | | | |
| | | | |
| o---------------------------------------o o-------------------o
|
| Figure 1. Manifold Of Connotative Impressions
¤~~~~~~~~~¤~~~~~~~~~¤~SUBALLYS~¤~~~~~~~~~¤~~~~~~~~~¤
References And Incidental Nuances (RAIN)
http://www-groups.dcs.st-andrews.ac.uk/~history/Mathematicians/Riemann.html
http://www.door.net/arisbe/menu/library/bycsp/newlist/nl-frame.htm
http://www.philosophy.ru/library/kant/01/cr_pure_reason.html
http://ez2www.com/go.php3?site=book&go=0387943382
http://hallmathematics.com/mathematics/1433.shtml
http://hallmathematics.com/mathematics/630.shtml
¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤~~~~~~~~~¤