Embedding#Riemannian geometry

In general topology, an embedding is a homeomorphism onto its image.{{harvnb|Hocking|Young|1988|page=73}}. {{harvnb|Sharpe|1997|page=16}}. More explicitly, an injective continuous map $f\; :\; X\; \backslash to\; Y$ between topological spaces $X$ and $Y$ is a topological embedding if $f$ yields a homeomorphism between $X$ and $f(X)$ (where $f(X)$ carries the subspace topology inherited from $Y$). Intuitively then, the embedding $f\; :\; X\; \backslash to\; Y$ lets us treat $X$ as a subspace of $Y$. Every embedding is injective and continuous. Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings that are neither open nor closed. The latter happens if the image $f(X)$ is neither an open set nor a closed set in $Y$.

For a given space $Y$, the existence of an embedding $X\; \backslash to\; Y$ is a topological invariant of $X$. This allows two spaces to be distinguished if one is able to be embedded in a space while the other is not.

If the domain of a function $f\; :\; X\; \backslash to\; Y$ is a topological space then the function is said to be {{visible anchor|locally injective at a point}} if there exists some neighborhood $U$ of this point such that the restriction $f\backslash big\backslash vert\_U\; :\; U\; \backslash to\; Y$ is injective. It is called {{visible anchor|locally injective}} if it is locally injective around every point of its domain. Similarly, a {{visible anchor|local topological embedding|text=local (topological, resp. smooth) embedding}} is a function for which every point in its domain has some neighborhood to which its restriction is a (topological, resp. smooth) embedding.

Every injective function is locally injective but not conversely. Local diffeomorphisms, local homeomorphisms, and smooth immersions are all locally injective functions that are not necessarily injective. The inverse function theorem gives a sufficient condition for a continuously differentiable function to be (among other things) locally injective. Every fiber of a locally injective function $f\; :\; X\; \backslash to\; Y$ is necessarily a discrete subspace of its domain $X.$

In differential topology:

Let $M$ and $N$ be smooth manifolds and $f:M\backslash to\; N$ be a smooth map. Then $f$ is called an immersion if its derivative is everywhere injective. An embedding, or a smooth embedding, is defined to be an immersion that is an embedding in the topological sense mentioned above (i.e. homeomorphism onto its image).{{harvnb|Bishop|Crittenden|1964|page=21}}. {{harvnb|Bishop|Goldberg|1968|page=40}}. {{harvnb|Crampin|Pirani|1994|page=243}}. {{harvnb|do Carmo|1994|page=11}}. {{harvnb|Flanders|1989|page=53}}. {{harvnb|Gallot|Hulin|Lafontaine|2004|page=12}}. {{harvnb|Kobayashi|Nomizu|1963|page=9}}. {{harvnb|Kosinski|2007|page=27}}. {{harvnb|Lang|1999|page=27}}. {{harvnb|Lee|1997|page=15}}. {{harvnb|Spivak|1999|page=49}}. {{harvnb|Warner|1983|page=22}}.

In other words, the domain of an embedding is diffeomorphic to its image, and in particular the image of an embedding must be a submanifold. An immersion is precisely a local embedding, i.e. for any point $x\backslash in\; M$ there is a neighborhood $x\backslash in\; U\backslash subset\; M$ such that $f:U\backslash to\; N$ is an embedding.

When the domain manifold is compact, the notion of a smooth embedding is equivalent to that of an injective immersion.

An important case is $N\; =\; \backslash mathbb\{R\}^n$. The interest here is in how large $n$ must be for an embedding, in terms of the dimension $m$ of $M$. The Whitney embedding theoremWhitney H., Differentiable manifolds, Ann. of Math. (2), 37 (1936), pp. 645–680 states that $n\; =\; 2m$ is enough, and is the best possible linear bound. For example, the real projective space $\backslash mathbb\{R\}\backslash mathrm\{P\}^m$ of dimension $m$, where $m$ is a power of two, requires $n\; =\; 2m$ for an embedding. However, this does not apply to immersions; for instance, $\backslash mathbb\{R\}\backslash mathrm\{P\}^2$ can be immersed in $\backslash mathbb\{R\}^3$ as is explicitly shown by Boy's surface—which has self-intersections. The Roman surface fails to be an immersion as it contains cross-caps.

{{Anchor|ProperEmbedding}}An embedding is proper if it behaves well with respect to boundaries: one requires the map $f:\; X\; \backslash rightarrow\; Y$ to be such that

• $f(\backslash partial\; X)\; =\; f(X)\; \backslash cap\; \backslash partial\; Y$, and
• $f(X)$ is transverse to $\backslash partial\; Y$ in any point of $f(\backslash partial\; X)$.

The first condition is equivalent to having $f(\backslash partial\; X)\; \backslash subseteq\; \backslash partial\; Y$ and $f(X\; \backslash setminus\; \backslash partial\; X)\; \backslash subseteq\; Y\; \backslash setminus\; \backslash partial\; Y$. The second condition, roughly speaking, says that $f(X)$ is not tangent to the boundary of $Y$.

In Riemannian geometry and pseudo-Riemannian geometry:

Let $(M,g)$ and $(N,h)$ be Riemannian manifolds or more generally pseudo-Riemannian manifolds.

An isometric embedding is a smooth embedding $f:M\backslash rightarrow\; N$ that preserves the (pseudo-)metric in the sense that $g$ is equal to the pullback of $h$ by $f$, i.e. $g=f^\{*\}h$. Explicitly, for any two tangent vectors $v,w\backslash in\; T\_x(M)$ we have

:$g(v,w)=h(df(v),df(w)).$

Analogously, isometric immersion is an immersion between (pseudo)-Riemannian manifolds that preserves the (pseudo)-Riemannian metrics.

Equivalently, in Riemannian geometry, an isometric embedding (immersion) is a smooth embedding (immersion) that preserves length of curves (cf. Nash embedding theorem).Nash J., The embedding problem for Riemannian manifolds, Ann. of Math. (2), 63 (1956), 20–63.

In general, for an algebraic category $C$, an embedding between two $C$-algebraic structures $X$ and $Y$ is a $C$-morphism {{nowrap|$e:X\backslash rightarrow\; Y$}} that is injective.

In field theory, an embedding of a field $E$ in a field $F$ is a ring homomorphism {{nowrap|$\backslash sigma:E\backslash rightarrow\; F$}}.

The kernel of $\backslash sigma$ is an ideal of $E$, which cannot be the whole field $E$, because of the condition {{nowrap|$1=\backslash sigma(1)=1$}}. Furthermore, any field has as ideals only the zero ideal and the whole field itself (because if there is any non-zero field element in an ideal, it is invertible, showing the ideal is the whole field). Therefore, the kernel is $0$, so any embedding of fields is a monomorphism. Hence, $E$ is isomorphic to the subfield $\backslash sigma(E)$ of $F$. This justifies the name embedding for an arbitrary homomorphism of fields.

{{further|Substructure (mathematics)|Elementary equivalence}}

If $\backslash sigma$ is a signature and $A,B$ are $\backslash sigma$-structures (also called $\backslash sigma$-algebras in universal algebra or models in model theory), then a map $h:A\; \backslash to\; B$ is a $\backslash sigma$-embedding exactly if all of the following hold:

• $h$ is injective,
• for every $n$-ary function symbol $f\; \backslash in\backslash sigma$ and $a\_1,\backslash ldots,a\_n\; \backslash in\; A^n,$ we have $h(f^A(a\_1,\backslash ldots,a\_n))=f^B(h(a\_1),\backslash ldots,h(a\_n))$,
• for every $n$-ary relation symbol $R\; \backslash in\backslash sigma$ and $a\_1,\backslash ldots,a\_n\; \backslash in\; A^n,$ we have $A\; \backslash models\; R(a\_1,\backslash ldots,a\_n)$ iff $B\; \backslash models\; R(h(a\_1),\backslash ldots,h(a\_n)).$

Here $A\backslash models\; R\; (a\_1,\backslash ldots,a\_n)$ is a model theoretical notation equivalent to $(a\_1,\backslash ldots,a\_n)\backslash in\; R^A$. In model theory there is also a stronger notion of elementary embedding.

In order theory, an embedding of partially ordered sets is a function $F$ between partially ordered sets $X$ and $Y$ such that

:$\backslash forall\; x\_1,x\_2\backslash in\; X:\; x\_1\backslash leq\; x\_2\; \backslash iff\; F(x\_1)\backslash leq\; F(x\_2).$

Injectivity of $F$

follows quickly from this definition. In domain theory, an additional requirement is that

:$\backslash forall\; y\backslash in\; Y:\backslash \{x\; \backslash mid\; F(x)\; \backslash leq\; y\backslash \}$ is directed.

A mapping $\backslash phi:\; X\; \backslash to\; Y$ of metric spaces is called an embedding

(with distortion $C>0$) if

:$L\; d\_X(x,\; y)\; \backslash leq\; d\_Y(\backslash phi(x),\; \backslash phi(y))\; \backslash leq\; CLd\_X(x,y)$

for every $x,y\backslash in\; X$ and some constant $L>0$.

An important special case is that of normed spaces; in this case it is natural to consider linear embeddings.

One of the basic questions that can be asked about a finite-dimensional normed space $(X,\; \backslash |\; \backslash cdot\; \backslash |)$ is, what is the maximal dimension $k$ such that the Hilbert space $\backslash ell\_2^k$ can be linearly embedded into $X$ with constant distortion?

The answer is given by Dvoretzky's theorem.

In category theory, there is no satisfactory and generally accepted definition of embeddings that is applicable in all categories. One would expect that all isomorphisms and all compositions of embeddings are embeddings, and that all embeddings are monomorphisms. Other typical requirements are: any extremal monomorphism is an embedding and embeddings are stable under pullbacks.

Ideally the class of all embedded subobjects of a given object, up to isomorphism, should also be small, and thus an ordered set. In this case, the category is said to be well powered with respect to the class of embeddings. This allows defining new local structures in the category (such as a closure operator).

In a concrete category, an embedding is a morphism $f:A\backslash rightarrow\; B$ that is an injective function from the underlying set of $A$ to the underlying set of $B$ and is also an initial morphism in the following sense:

If $g$ is a function from the underlying set of an object $C$ to the underlying set of $A$, and if its composition with $f$ is a morphism $fg:C\backslash rightarrow\; B$, then $g$ itself is a morphism.

A factorization system for a category also gives rise to a notion of embedding. If $(E,M)$ is a factorization system, then the morphisms in $M$ may be regarded as the embeddings, especially when the category is well powered with respect to $M$. Concrete theories often have a factorization system in which $M$ consists of the embeddings in the previous sense. This is the case of the majority of the examples given in this article.

As usual in category theory, there is a dual concept, known as quotient. All the preceding properties can be dualized.

An embedding can also refer to an embedding functor.

• Embedding (machine learning)
• Ambient space
• Closed immersion
• Cover
• Dimensionality reduction
• Flat (geometry)
• Immersion
• Johnson–Lindenstrauss lemma
• Submanifold
• Subspace
• Universal space

{{reflist}}

{{refbegin}}

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• {{cite book|last1=Bishop|first1=Richard Lawrence|author1-link=Richard L. Bishop|last2=Goldberg|first2=Samuel Irving|title=Tensor Analysis on Manifolds|publisher=The Macmillan Company|year=1968|edition=First Dover 1980|isbn=0-486-64039-6|url=https://archive.org/details/tensoranalysison00bish}}
• {{cite book|last1=Crampin|first1=Michael|last2=Pirani|first2=Felix Arnold Edward|author-link2=Felix Pirani|title=Applicable differential geometry|publisher=Cambridge University Press|location=Cambridge, England|year=1994|isbn=978-0-521-23190-9|url-access=registration|url=https://archive.org/details/applicablediffer0000cram}}
• {{cite book|title = Riemannian Geometry|first=Manfredo Perdigao | last = do Carmo |author-link=Manfredo do Carmo | year = 1994|publisher=Birkhäuser Boston |isbn=978-0-8176-3490-2}}
• {{cite book|last=Flanders|first=Harley|author-link=Harley Flanders|title=Differential forms with applications to the physical sciences|publisher=Dover|year=1989|isbn=978-0-486-66169-8}}
• {{Cite book| last1=Gallot | first1=Sylvestre | author1-link=Sylvestre Gallot | last2=Hulin | first2=Dominique |author2-link=Dominique Hulin| last3=Lafontaine | first3=Jacques | title=Riemannian Geometry | publisher=Springer-Verlag | location=Berlin, New York | edition=3rd | isbn=978-3-540-20493-0 | year=2004}}
• {{cite book|first1=John Gilbert|last1=Hocking|first2=Gail Sellers|last2=Young|title=Topology|year=1988|orig-year=1961|publisher=Dover|isbn=0-486-65676-4|url=https://archive.org/details/topology00hock_0}}
• {{cite book|last=Kosinski|first=Antoni Albert|year=2007|orig-year=1993|title=Differential manifolds|location=Mineola, New York|publisher=Dover Publications|isbn=978-0-486-46244-8}}
• {{Cite book| isbn = 978-0-387-98593-0 | title = Fundamentals of Differential Geometry | last1 = Lang | first1 = Serge |author-link1=Serge Lang| year = 1999 |publisher=Springer|location=New York| series = Graduate Texts in Mathematics}}
• {{cite book|last1=Kobayashi|first1=Shoshichi|author-link1=Shoshichi Kobayashi|last2=Nomizu|first2=Katsumi|author-link2=Katsumi Nomizu| title = Foundations of Differential Geometry, Volume 1| publisher=Wiley-Interscience |location=New York| year=1963}}
• {{cite book|first=John Marshall|last=Lee|authorlink = John M. Lee|title=Riemannian manifolds|publisher=Springer Verlag|year=1997|isbn=978-0-387-98322-6}}
• {{cite book| first = R.W. | last = Sharpe | title = Differential Geometry: Cartan's Generalization of Klein's Erlangen Program | publisher = Springer-Verlag, New York | year = 1997| isbn = 0-387-94732-9}}.
• {{cite book|last=Spivak|first=Michael|author-link=Michael Spivak|title=A Comprehensive introduction to differential geometry (Volume 1)|year=1999|orig-year=1970|publisher=Publish or Perish|isbn=0-914098-70-5}}
• {{cite book| first=Frank Wilson| last = Warner |authorlink = Frank Wilson Warner| title = Foundations of Differentiable Manifolds and Lie Groups | publisher = Springer-Verlag, New York | year = 1983| isbn = 0-387-90894-3}}.

{{refend}}

• {{cite book|last=Adámek|first=Jiří|author2=Horst Herrlich |author3=George Strecker |title=Abstract and Concrete Categories (The Joy of Cats)|url=http://katmat.math.uni-bremen.de/acc/|year=2006}}
• [http://www.map.mpim-bonn.mpg.de/Embedding Embedding of manifolds] on the Manifold Atlas

{{set index article}}

Category:Abstract algebra

Category:Category theory

Category:General topology

Category:Differential topology

Category:Functions and mappings

Category:Maps of manifolds

Category:Model theory

Category:Order theory