# Differentiable manifold

A locally Euclidean space with a differentiable structure. Let $X$ be a topological Hausdorff space. $X$ is known as a locally Euclidean space or as a topological manifold of dimension $n$ if for each point $x \in X$ a neighbourhood $U$ of $x$ can be found that is homeomorphic to an open set of $\mathbf R ^ {n}$. The pair $( U , \phi )$, where $\phi$ is this homeomorphism, is known as a local chart of $X$ at $x$. Thus, to each point corresponds a selection of $n$ real numbers $( x ^ {1} \dots x ^ {n} )$, known as the coordinates of $x$ in the chart $( U , \phi )$.

A family of charts $\{ ( U _ \alpha , \phi _ \alpha ) \}$, $\alpha \in A$, is known as an $n$-dimensional $C ^ {k}$-atlas $( 0 \leq k \leq \infty , a )$ of $X$ if a) the totality of all $U _ \alpha$ covers $X$, $X = \cup _ {\alpha \in A } U _ \alpha$; and b) for any $\alpha , \beta \in A$ such that $U _ \alpha \cap U _ \beta \neq \emptyset$, the mapping

$$\phi _ \beta ^ \alpha = \phi _ \beta \circ \phi _ \alpha ^ {- 1} : \ \phi _ \alpha ( U _ \alpha \cap U _ \beta ) \rightarrow \phi _ \beta ( U _ \alpha \cap U _ \beta )$$

belongs to the class of differentiability $C ^ {k}$; $\phi _ \beta ^ \alpha$ is a differentiable mapping with non-vanishing Jacobian and is known as a transformation of coordinates $x$ from the chart $( U _ \alpha , \phi _ \alpha )$ into the chart $( U _ \beta , \phi _ \beta )$.

Two $C ^ {k}$-atlases are said to be equivalent if their union is again a $C ^ {k}$-atlas. The set of $C ^ {k}$-atlases is thus subdivided into equivalence classes, known as $C ^ {k}$- structures; if $1 \leq k \leq \infty$, they are known as differentiable (or smooth) structures, while if $k = a$ they are known as analytic structures. The topological manifold $X$ with a $C ^ {k}$-structure is known as a $C ^ {k}$-manifold, or as a differentiable manifold of class $C ^ {k}$.

The concept of a differentiable structure may be introduced for an arbitrary set $X$ by replacing the homeomorphisms $\phi _ \alpha$ by bijective mappings on open sets of $\mathbf R ^ {n}$; here, the topology of the $C ^ {k}$-manifold is described as the topology of the union, constructed from an arbitrary atlas of the corresponding structure. In such a case $n$- dimensional manifolds clearly have an $n$-dimensional $C ^ {0}$-structure.

Problems of analytical and algebraic geometry make it necessary to consider in the definition of a differentiable structure not only the space $\mathbf R ^ {n}$, but also more general spaces, such as $\mathbf C ^ {n}$ or even $K ^ {n}$ where $K$ is a complete non-discretely normed field. Thus, if $K = \mathbf C$, the corresponding $C ^ {k}$-structure, $k \geq 1$, invariably proves to be a $C ^ {a}$-structure and is called complex-analytic, or simply complex, while the corresponding differentiable manifold is known as a complex manifold. Such a manifold also carries a natural real $C ^ {a}$-structure.

Any $C ^ {a}$-manifold contains a $C ^ \infty$-structure, and there is a $C ^ {r}$-structure on a $C ^ {k}$- manifold, $0 \leq k \leq \infty$, if $0 \leq r \leq k$. Conversely, any paracompact $C ^ {r}$-manifold, $r \geq 1$, may be provided with a $C ^ {a}$-structure compatible with the given one, and this structure is unique, up to an isomorphism (see below). It may happen, however, that a $C ^ {0}$-manifold cannot be provided with a $C ^ {1}$-structure (i.e. there exist non-smoothable manifolds, cf. Non-smoothable manifold), and even if it can be provided with such a structure, the structure need not be unique. For example, the number $\theta ( n)$ of $C ^ {1}$ non-isomorphic $C ^ \infty$ structures on the $n$-dimensional sphere is:

<tbody> </tbody>
 $n$ 1 2 3 4 5 6 7 8 9 10 11 12 $\theta ( n)$ 1 1 1 ? 1 1 28 2 8 6 992 1

Let $f : X \rightarrow Y$ be a continuous mapping of $C ^ {r}$-manifolds $X , Y$; it is known as a $C ^ {k}$-morphism (or as a $C ^ {k}$-mapping, $k \leq r$, or as a mapping of class $C ^ {k}$) of differentiable manifolds if for any pair of charts $( U _ \alpha , \phi _ \alpha )$ on $X$ and $( V _ \beta , \psi _ \beta )$ on $Y$ such that $f ( U _ \alpha ) \subset V _ \beta$, the mapping

$$\psi _ \alpha \circ f \circ \phi _ \beta ^ {- 1} : \phi _ \alpha ( U _ \alpha ) \rightarrow \psi _ \beta ( V _ \beta )$$

belongs to the class $C ^ {k}$. A bijective mapping $f$ such that it and $f ^ { - 1 }$ are $C ^ {n}$-mappings is called a $C ^ {n}$-isomorphism (or a diffeomorphism of class $C ^ {n}$). In such a case $X$ and $Y$ and their determining $C ^ {r}$-structures are said to be $C ^ {n}$-diffeomorphic.

A subspace $Y$ of an $n$-dimensional $C ^ {k}$-manifold $X$ is called a $C ^ {k}$-submanifold of dimension $m$ in $X$ if for any point $y \in Y$ there exists a neighbourhood $V \subset Y$ of it and a chart $( U , \phi )$ of the $C ^ {k}$-structure $X$ such that $V \subset U$ and $\phi$ induces a homeomorphism of $V$ onto the intersection of $\phi ( U \cap Y )$ with the closed subspace $\mathbf R ^ {m} \subset \mathbf R ^ {n}$; in other words, there exists a chart with coordinates $x ^ {1} \dots x ^ {n}$ such that $U \cap Y$ is defined by the relations $x ^ {m+ 1} = \dots = x ^ {n} = 0$.

A mapping $f : X \rightarrow Y$ is said to be a $C ^ {k}$-imbedding if $f( X )$ is a $C ^ {k}$-submanifold in $Y$ and if $X \rightarrow f ( X )$ is a $C ^ {k}$-diffeomorphism. Any $n$-dimensional $C ^ {k}$-manifold permits an imbedding in $\mathbf R ^ {2n+} 1$ and even in $\mathbf R ^ {2n}$. Moreover, the set of such imbeddings is everywhere dense in the space of mappings $C ^ {k} ( X , \mathbf R ^ {2n+ 1} )$ with respect to the compact-open topology. Thus, regarding a differentiable manifold as a submanifold of a Euclidean space is one of the ways of interpreting the theory of differentiable manifolds; for example, the above theorems on $C ^ {a}$-structures can be proved in this manner.

There are two fundamental problems in the topology of differentiable manifolds (which is also referred to as differential topology). The first problem is the classification of differentiable manifolds. There exist three main classes of differentiable manifolds — closed (or compact) manifolds, compact manifolds with boundary and open manifolds. Important invariants by which differentiable manifolds are distinguished are the homotopy type and the tangent bundle, in particular the characteristic classes (cf. Characteristic class). Using these a classification of smooth structures for simply-connected manifolds of given homotopy type has been given. Another invariant — the bordism class of a differentiable manifold — was used in solving the generalized Poincaré conjecture, in the study of fixed points under the action of a group on a manifold, etc. This involved the introduction of differentiable structures on manifolds with boundary and of a smoothing apparatus. Finally, methods of algebraic topology also proved useful in this context, since, for example, they permitted to establish that any $C ^ {1}$-manifold can be triangulated.

The second problem is the classification of mappings of differentiable manifolds. The first class to be considered are immersions, which are a generalization of imbeddings; their classification is reduced to a homotopy problem, as distinct from imbeddings, which have not yet (1987) been completely classified (cf. Topology of imbeddings), and submersions, or fibrations, of one differentiable manifold into another. In particular, the concept of a transversal mapping along a submanifold plays an important role in problems of stability and in the study of typical singularities of mappings. The existence of transversal mappings is ensured by theorems such as Sard's theorem (cf. Sard theorem). All this, and problems in differential dynamics, dealing with the structure of various groups of diffeomorphisms (cf. Diffeomorphism), in particular of integral trajectories and singular points of vector fields on differentiable manifolds (dynamical systems), as well as the various equivalence relationships — isotopy, topological and $C ^ {k}$-conjugacy, etc. — makes it necessary to study finite-dimensional spaces $\mathbf R ^ {n}$ together with arbitrary Banach (or Hilbert) spaces and to determine corresponding differentiable structures. This implies finding additional conditions that are reasonable from the point of view of applications, e.g., a differentiable manifold is separable if and only if the coordinate transformations have a closed graph. In general, infinite-dimensional manifolds provided with such a structure — known as Banach or Hilbert manifolds, respectively, manifolds of mappings of finite-dimensional manifolds being their typical example — are a useful outcome of studies and geometrical interpretation of problems of approximation of mappings (as in the imbedding theorem above), in the analysis of loop spaces (a suitable domain for the construction of Morse theory, cf. Loop space), etc.

Differentiable manifolds form a natural base for developing differential geometry. Supplementary infinitesimal structures — orientation, metric, connections, etc. — are introduced on differentiable manifolds, after which a study is made of the objects which are invariant with respect to the group of diffeomorphisms which preserve the supplementary structure. Conversely, the use of a specific structure permits one to study the structure of the differential manifold itself. The simplest example is the expression of the characteristic classes in terms of the curvature of a differentiable manifold with a linear connection.

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