Differential geometry

As a branch of mathematics, differential geometry represents the synthesis of analysis and geometry .

Historical development and current areas of application

A number of fundamental works on differential geometry come from Carl Friedrich Gauß . At that time, mathematics was still closely linked to various fields of application. This theory delivered important results in the fields of cartography , navigation and geodesy . Among other things, the theory of map projection developed , from which the terms geodetic line and Gaussian curvature come. In addition, CF Gauss already asked himself whether the sum of the angles of a very large triangle measured by bearing actually amounts to exactly 180 degrees, and thus proves to be a pioneer of modern differential geometry.

Modern differential geometry is mainly used in general relativity and in satellite navigation . It enables the description of phenomena such as the astronomical deflection of light or the rotation of the perihelion of Mercury , which can be confirmed by experiments or observation . In the theory of relativity, coordinate transformations correspond to the change of reference systems from which a phenomenon is observed. This corresponds to different states of motion of the measuring apparatus or the observer.

Another important area of ​​application is in materials science in the theory of defects and plasticity .

Sub-areas

Elementary differential geometry

The first work on differential geometry deals with both curves and two-dimensional curved surfaces in three-dimensional real visual space . From a historical point of view, Gauss's work made it possible for the first time to quantitatively record the curvature of the two-dimensional surface of a sphere, for example.

Another motivation for the development of elementary differential geometry came from the mathematical problem of minimal surfaces . The soap skins that occur in nature can be described as minimal areas. The shape or mathematical representation of these surfaces can be developed using the methods from the calculus of variations . The geometric properties of these surfaces such as curvature or distances between any points on a minimal surface, on the other hand, are more likely to be calculated using the methods of differential geometry.

Differential topology

→ Main article : differential topology

The differential topology is the basis for most of the modern areas of differential geometry. In contrast to elementary differential geometry, the geometric objects are described intrinsically in differential topology , that is, the objects are defined without recourse to a surrounding space. The central concept is that of the differentiable manifold : A -dimensional manifold is a geometric object (more precisely: a topological space ) that looks locally like the -dimensional real space. The classic example that also motivates the terminology is the earth's surface. It can be described in small excerpts using maps , that is, small parts “look like” the plane. However, the entire surface of the earth cannot be identified with the plane. In addition, differentiable manifolds have a structure that allows us to speak of differentiable functions. This differentiable structure makes it possible to use local analytical methods in the maps. In addition, one can examine the manifold globally as a topological space. The differential topology tries to establish connections between the local analytical and the global topological properties. An example of such a connection is de Rham's theorem . ${\displaystyle n}$ ${\displaystyle n}$

Riemannian geometry

→ Main article : Riemannian geometry

There is no predefined length measurement on a differentiable manifold. If it is given as an additional structure, one speaks of Riemannian manifolds . These manifolds are the subject of Riemannian geometry, which also examines the associated concepts of curvature , the covariant derivative and the parallel transport on these sets. These terms can, however, also be defined for “non-Riemannian” or “non-pseudoriemannian” spaces and only require the general differential-geometric concept of the context (more precisely: general affine differential geometry in contrast to metric differential geometry, see below.)

Semi-Riemannian differential geometry

If, instead of the positive-definite metric of a Riemannian manifold, a non-definite metric is assumed (given by a non-definite Hermitian or symmetrically-non-definite non -degenerate bilinear form ), a semi- or pseudo-Riemannian manifold is obtained . The Lorentzian manifolds of the general theory of relativity are a special case .

Finsler's geometry

The subject of Finsler's geometry are the Finsler's manifolds , i.e. manifolds whose tangent space is equipped with a Banach norm, i.e. a mapping with the following properties: ${\displaystyle F\colon TM\to [0,\infty )}$

1. ${\displaystyle F(rX)=|r|F(X)\,}$, for and , ${\displaystyle X\in TM}$ ${\displaystyle r\in \mathbb {R} }$
2. ${\displaystyle F(X+Y)\leq F(X)+F(Y),}$
3. ${\displaystyle F}$is smooth on , ${\displaystyle TM\setminus 0}$
4. the vertical Hessian matrix is positive definite .

Finsler's manifolds also play a role in theoretical physics as more general candidates for the structural description of spacetime.

Symplectic geometry

→ Main article : Symplectic geometry

Instead of a symmetric nondegenerate bilinear form, an antisymmetric nondegenerate bilinear form ω is given. If this is also closed, i.e. d ω = 0, one speaks of a symplectic manifold. Because a symplectic vector space necessarily has an even dimension, symplectic manifolds also have an even dimension. The first important finding is Darboux's theorem , according to which symplectic manifolds are locally isomorphic to T ^* R ⁿ . In contrast to semi-Riemannian manifolds, there are no (non-trivial) local symplectic invariants (apart from the dimension), but only global symplectic invariants. The Poisson manifolds , which do not have a bilinear form but only an antisymmetric bivector , also count as a generalization . This induces a Lie bracket between the functions. Symplectic geometry is used in Hamiltonian mechanics , a branch of theoretical mechanics .

Contact geometry

→ Main article : Contact geometry

The analogue to symplectic geometry for odd-dimensional manifolds is contact geometry. A contact structure on a -dimensional manifold is a family of hyperplanes of the tangential bundle that are maximally non-integrable. Locally, these hyperplanes can be represented as a core of a 1-form , i.e. H. ${\displaystyle (2n+1)}$ ${\displaystyle M}$ ${\displaystyle H}$ ${\displaystyle \alpha }$

${\displaystyle H_{p}=\ker \alpha _{p}\subset T_{p}M}$.

Conversely, a form of contact is locally uniquely determined by the family , except for one non-vanishing factor. The non-integrability means that dα is non-degenerate restricted to the hyperplane. If the family can be described globally by a 1-form , then contact form is if and only if ${\displaystyle H}$ ${\displaystyle H}$ ${\displaystyle \alpha }$ ${\displaystyle \alpha }$

${\displaystyle \alpha \wedge (d\alpha )^{n}}$is a volume shape . ${\displaystyle M}$

A theorem analogous to Darboux's theorem for symplectic manifolds applies, namely that all contact manifolds of the dimension are locally isomorphic. This means that there are only global invariants in contact geometry. ${\displaystyle 2n+1}$

Complex geometry and Kähler geometry

→ Main article : Complex geometry

Complex geometry is the study of complex manifolds, that is, manifolds that look like locally and whose transition functions are complex-differentiable (holomorphic). Because of the analytical properties of complex-differentiable functions, one often has uniqueness properties of the continuation of local functions / vector fields. That is why one is mostly dependent on the theory of sheaves in global studies . An almost-complex structure on a smooth manifold is a map such that . Thus all almost complex manifolds are of even dimension. The difference between an almost-complex and a complex manifold is the integrability of the almost-complex structure. This is measured by the Nijenhuis tensor . ${\displaystyle \mathbb {C} ^{n}}$ ${\displaystyle J\colon TM\to TM}$ ${\displaystyle J^{2}=-1}$ ${\displaystyle N_{J}}$

A Hermitian manifold is a complex manifold with a Hermitian metric on the complexified real tangent bundle. In particular, must be compatible with the complex structure , namely ${\displaystyle g}$ ${\displaystyle g}$ ${\displaystyle J}$

${\displaystyle g(X,Y)=g(JX,JY)}$for everyone . ${\displaystyle X,Y\in T_{x}M}$

Hermitian manifolds, whose Hermitian metrics are additionally compatible with a symplectic form, have proven to be particularly structurally rich. H.

${\displaystyle g(JX,Y)=\omega (X,Y)}$with . ${\displaystyle d\omega =0}$

In this case one speaks of a Kahler manifold .

Finally, Cauchy-Riemann geometry deals with bounded complex manifolds.

Lie group theory

Just as groups are based on sets , manifolds are the basis of Lie groups . The Lie groups named after Sophus Lie appear in many places in mathematics and physics as continuous symmetry groups, for example as groups of rotations of space. The study of the transformation behavior of functions under symmetries leads to the representation theory of Lie groups.

Global Analysis

Global analysis is also a branch of differential geometry that is closely related to topology. Sometimes the sub-area is also called analysis on manifolds. In this mathematical research area, ordinary and partial differential equations are investigated on differentiable manifolds. In this theory, local methods from functional analysis , micro- local analysis and the theory of partial differential equation and global methods from geometry and topology are used. Since this mathematical sub-area uses many methods of analysis in comparison to the other sub-areas of differential geometry, it is partly understood as a sub-area of ​​analysis.

Even the first work on differential equations contained aspects of global analysis. The studies of George David Birkhoff in the field of dynamic systems and the theory of geodesics by Harold Calvin Marston Morse are early examples of methods of global analysis. Central results of this mathematical sub-area are the work of Michael Francis Atiyah , Isadore M. Singer and Raoul Bott . Particularly noteworthy here are the Atiyah-Singer index theorem and the Atiyah-Bott fixed-point theorem , which is a generalization of Lefschetz's fixed-point theorem from topology.

Methods

Coordinate transformations

Coordinate transformations are an important tool in differential geometry to enable the adaptation of a problem to geometric objects. If, for example, distances on a spherical surface are to be examined, spherical coordinates are mostly used. If one looks at Euclidean distances in space, however, one uses more Cartesian coordinates. From a mathematical point of view, it should be noted that coordinate transformations are always bijective , as often as desired, continuously differentiable maps. The inverse of the coordinate transformation under consideration always also exists.

A simple example is the transition from Cartesian coordinates in the plane to polar coordinates . Each position vector of two-dimensional Euclidean space can be expressed in this representation by the coordinates and in the following way ${\displaystyle r\in [0,\infty [}$ ${\displaystyle \phi \in [0,2\pi [}$

${\displaystyle {\vec {\mathbf {r} }}={\begin{pmatrix}x\\y\end{pmatrix}}={\vec {f}}(r,\phi )={\begin{pmatrix}r\,\cos \phi \\r\,\sin \phi \end{pmatrix}}}$

${\displaystyle x}$and are also referred to as component functions of. They are calculated as a function of the two coordinates : ${\displaystyle y}$ ${\displaystyle f}$ ${\displaystyle (r,\phi )}$

${\displaystyle x(r,\phi )=r\,\cos \phi \,,\,\,\,\,y(r,\phi )=r\,\sin \phi \,\,.}$

If, in general, all coordinates of the new coordinate system are kept constant except for one coordinate and the individual coordinates are changed within the definition range, lines are created in Euclidean space which are also referred to as coordinate lines. In the case of the specified polar coordinates, concentric circles with a radius around the coordinate origin of the Euclidean coordinate system are created with a constant coordinate. With a constant coordinate, half-lines are created that start in the coordinate origin of the Euclidean coordinate system and then run. With the help of these coordinate lines, a new, spatially rotated and again right-angled coordinate system can be defined in an obvious way for every point in Euclidean space. For this reason, polar coordinates are also referred to as right-angled coordinates. The axes of the rotated coordinate system are precisely the tangents to the coordinate lines that run through the point . The base vectors of these position-dependent and right-angled coordinate systems can be calculated directly from the partial derivatives of the position vector according to the above-mentioned representation according to the variable coordinates . The total differentials of the position vector can also be given via the partial derivatives: ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle (x,y)=(0,0)}$ ${\displaystyle \phi }$ ${\displaystyle r\rightarrow \infty }$ ${\displaystyle P\in \mathbb {R} ^{2}}$ ${\displaystyle P}$ ${\displaystyle (r,\phi )}$

${\displaystyle \mathrm {d} x={\frac {\partial x}{\partial r}}\mathrm {d} r+{\frac {\partial x}{\partial \phi }}\mathrm {d} \phi =\cos \phi \,\mathrm {d} r-r\cdot \sin \phi \,\mathrm {d} \phi }$

${\displaystyle \mathrm {d} y={\frac {\partial y}{\partial r}}\mathrm {d} r+{\frac {\partial y}{\partial \phi }}\mathrm {d} \phi =\sin \phi \,\mathrm {d} r+r\cdot \cos \phi \,\mathrm {d} \phi }$

The differentials are also referred to as coordinate differentials . In this example, the infinitesimal quantities linked with the differential operator “ ” do not always have the meaning of a distance. It is rather easy to show that for the distances in the radial or azimuthal direction it is true that it     is, but ; ie only with the prefactor " " results from integration over from 0 to a known quantity of the dimension "length", namely the circumference . ${\displaystyle \mathrm {d} x,\mathrm {d} y,\mathrm {d} r,\mathrm {d} \phi }$ ${\displaystyle \mathrm {d} }$ ${\displaystyle \mathrm {d} l_{r}\,:=\mathrm {d} r}$ ${\displaystyle \mathrm {d} l_{\phi }:=r\cdot \mathrm {d} \phi }$ ${\displaystyle r}$ ${\displaystyle \mathrm {d} \Phi }$ ${\displaystyle 2\pi }$ ${\displaystyle r\cdot 2\pi }$

The polar coordinates or their three-dimensional generalization, the spherical coordinates, are also referred to as curvilinear as they facilitate the calculation of the distance on a curved surface, e.g. B. the spherical surface, allow. As with other standard examples, such as the cylindrical coordinates , the elliptical coordinates , etc., these are orthogonal, curvilinear coordinates (see also: Curvilinear coordinates ).

An essential tool of classical differential geometry are coordinate transformations between any coordinates in order to be able to describe geometric structures.

The differential operators formed with magnitude , known from analysis, can be extended relatively easily to orthogonal curvilinear differential operators. For example, in general orthogonal curvilinear coordinates when using three parameters and the associated unit vectors in the direction of, the following relationships apply with quantities that are not necessarily constant, but can depend on , and : ${\displaystyle \nabla }$ ${\displaystyle u_{i},\,\,i=1,\dots ,3}$ ${\displaystyle \mathbf {e} _{i}}$ ${\displaystyle {\tfrac {\partial \mathbf {r} }{\partial u_{i}}}}$ ${\displaystyle a_{i}}$ ${\displaystyle u_{1}}$ ${\displaystyle u_{2}}$ ${\displaystyle u_{3}}$

${\displaystyle {\begin{aligned}\mathrm {d} \mathbf {r} &=\sum \limits _{i=1}^{3}{\rm {d}}l_{i}\mathbf {e} _{i}=\sum \limits _{i=1}^{3}\,a_{i}\,\mathrm {d} u_{i}\,\mathbf {e} _{i}\\{\rm {d}}V&=a_{1}\mathrm {d} u_{1}\cdot a_{2}\mathrm {d} u_{2}\cdot a_{3}\mathrm {d} u_{3}\\\nabla ^{2}f&={\frac {1}{a_{1}a_{2}a_{3}}}{\frac {\partial }{\partial u_{1}}}\left({\frac {a_{2}a_{3}}{a_{1}}}{\frac {\partial f}{\partial u_{1}}}\right)+\cdots +\cdots \end{aligned}}}$

The two further terms from the first term, indicated by dots, are created by cyclically interchanging the indices. denotes the Laplace operator . It can be composed of the scalar-valued div operator and the vector-valued grad operator according to ${\displaystyle \nabla ^{2}}$

${\displaystyle \nabla ^{2}f=\operatorname {div} (\operatorname {grad} f)}$

in which

${\displaystyle {\begin{aligned}\operatorname {div} \mathbf {v} \,&=\,{\frac {1}{a_{1}a_{2}a_{3}}}{\frac {\partial (a_{2}a_{3}v_{1})}{\partial u_{1}}}+\cdots +\cdots \\\operatorname {grad} f&=\sum \limits _{i=1}^{3}\,{\frac {1}{a_{i}}}{\frac {\partial f}{\partial u_{i}}}\,\mathbf {e} _{i}\end{aligned}}}$

The formula for the divergence is based on the coordinate-independent representation

${\displaystyle \operatorname {div} \mathbf {v} =\lim _{\Delta V\to 0}\,{\Big \{}{\frac {1}{|\Delta V|}}\,\iint \limits _{\partial (\Delta V)}\mathbf {v} \cdot \mathbf {n} {\rm {d}}^{(2)}A{\Big \}}\,\,,}$

integrating over the closed, bordering surface. denotes the associated outer normal vector, the corresponding infinitesimal surface element . In the most general case - i.e. for non-orthogonal, curvilinear coordinates - this formula can also be used. ${\displaystyle \Delta V}$ ${\displaystyle \mathbf {n} }$ ${\displaystyle {\rm {d}}^{(2)}A}$ ${\displaystyle \Delta V\to {\rm {d}}V}$

Covariant derivative

→ Main article : Relationship (differential geometry)

General derivative operators based on not necessarily orthogonal curvilinear coordinates are e.g. B. the covariant derivatives that u. a. be used in Riemannian spaces , where they are specifically related to the “inner product”, i.e. H. on the so-called " metric fundamental form " of the room. In other cases, however, they are independent of the existence of a local metric or can even be specified externally, e.g. B. in manifolds "with connection".

They enable u. a. the definition of connecting lines in curved spaces, e.g. B. the definition of geodesics in Riemannian space. Geodetic lines are the locally shortest connections between two points in these spaces. The longitudes on a sphere are examples of geodetic lines, but not the latitudes (exception: equator).

With the help of general coordinate transformations, the Christoffels symbols are defined in Riemannian space (and more generally in differential geometries “with a given context ”) . In accordance with the basic definition given below, these are explicitly included in the calculation of the covariant derivative of a vector field . ${\displaystyle \Gamma _{\alpha \beta }^{\mu }}$

The covariant derivative is a generalization of the partial derivative of flat (Euclidean) space for curved spaces. In contrast to the partial derivative , it has the tensor property; in Euclidean space it is reduced to a partial derivative. In curved space, the covariant derivatives of a vector field are generally not interchangeable with one another; their non-interchangeability is used to define the Riemann curvature tensor .

Another important term in connection with curved spaces is parallel translation . The covariant derivative of the components of a vector is zero with parallel shift. Nevertheless, the parallel displacement of a vector along a closed curve in curved space can lead to the displaced vector not coinciding with its starting vector.

The associated formalism is based on the rule that vectors are written as a sum , where u. U. (namely just in the above "parallel transport") are not the components , but only the basic elements of change, though gradually the obvious rule .  Covariant   and   partial   derivative, usually written with a semicolon or comma, are different, namely: ${\displaystyle \mathbf {v} }$ ${\displaystyle v^{\alpha }\mathbf {e} _{\alpha }}$ ${\displaystyle \,v^{\alpha }}$ ${\displaystyle \mathbf {e} _{\alpha }}$ ${\displaystyle d\mathbf {e} _{\alpha }=\Gamma _{\alpha \beta }^{\mu }dx^{\beta }\mathbf {e} _{\mu }}$

${\displaystyle v_{\,;\beta }^{\mu }\,dx^{\beta }=v_{\,,\beta }^{\mu }\,dx^{\beta }+\Gamma _{\alpha \beta }^{\mu }v^{\alpha }dx^{\beta }\,,}$  so     or also ${\displaystyle v_{\,;\beta }^{\mu }\,:=\,v_{\,,\beta }^{\mu }+\Gamma _{\alpha \beta }^{\mu }v^{\alpha }}$ ${\displaystyle \nabla _{\,\beta }v^{\mu }\,:=\,\partial _{\,\beta }v^{\mu }+\Gamma _{\alpha \beta }^{\mu }v^{\alpha }\,.}$

In manifolds with an additional structure (e.g. in Riemannian manifolds or in the so-called gauge theories ) this structure must of course be compatible with the transfer. This results in additional relationships for the Christoffel symbols. For example, in Riemannian spaces, the distance and angle relationships between two vectors must not change in the event of a parallel shift, and the Christoffel symbols are therefore calculated in a certain way from the metric structure alone.

Curvature tensor

The above-mentioned curvature of space results in an analogous way: If one shifts the basis vector in the mathematically positive sense (counterclockwise) first an infinitesimal segment in -direction and then an infinitesimal segment in -direction, one obtains a result that we can write in the form . If the order is reversed, i.e. if the direction of rotation is opposite, the result is the opposite. The difference can be written in the following form with a quantity that results from the Christoffel symbols: ${\displaystyle \mathbf {e} _{\alpha }}$ ${\displaystyle {\rm {d}}x^{\beta }}$ ${\displaystyle \beta }$ ${\displaystyle {\rm {d}}x^{\gamma }}$ ${\displaystyle \gamma }$ ${\displaystyle {\tfrac {1}{2}}\Delta ^{(2)}K_{\alpha }}$ ${\displaystyle \,\Delta ^{(2)}K_{\alpha }}$ ${\displaystyle R_{\;\;\alpha \beta \gamma }^{\lambda }}$

${\displaystyle {\begin{aligned}\Delta ^{(2)}K_{\alpha }&=R_{\;\;\alpha \beta \gamma }^{\lambda }{\rm {d}}x^{\beta }\,{\rm {d}}x^{\gamma \,}\mathbf {e} _{\lambda }\\&\equiv \,(\Gamma _{\alpha \beta }^{\mu }\,\Gamma _{\mu \gamma }^{\lambda }-\Gamma _{\alpha \gamma }^{\mu }\Gamma _{\mu \beta }^{\lambda }+\partial _{\gamma }\Gamma _{\alpha \beta }^{\lambda }-\partial _{\beta }\Gamma _{\gamma \alpha }^{\lambda }\,)\,{\rm {d}}x^{\beta }\,{\rm {d}}x^{\gamma \,}\,\mathbf {e} _{\lambda }\end{aligned}}}$

If the vector is shifted in parallel, the following results accordingly:    The components form the curvature tensor , a vector-valued differential form. (In the so-called Yang-Mills theories , this term is generalized by replacing, for example, "vector-valued" with Lie algebra-valued; see also Chern classes .) ${\displaystyle \mathbf {v} }$ ${\displaystyle v^{\lambda }\to v^{\lambda }\,\pm {\tfrac {1}{2}}\,R_{\;\;\alpha \beta \gamma }^{\lambda }\,v^{\alpha }\,{\rm {d}}x^{\beta }\,{\rm {d}}x^{\gamma }\,.}$ ${\displaystyle R_{\;\;\alpha \beta \gamma }^{\lambda }}$

In particular, the existence of the curvature tensor does not presuppose that one is dealing with metric or pseudometric spaces as in physics (see above), but only the affinity is assumed for the structure of the transmission .

literature

Elementary differential geometry

• W. Blaschke , K. Leichtweiß : Elementary differential geometry. (= Lectures on differential geometry. 1 = The basic teachings of the mathematical sciences in individual representations. 1). 5th, completely revised edition. Springer-Verlag, Berlin et al. 1973, ISBN 3-540-05889-3 .
• Manfredo P. do Carmo: Differential geometry of curves and surfaces (= Vieweg studies. Advanced course in mathematics. 55). Vieweg & Sohn, Braunschweig et al. 1983, ISBN 3-528-07255-5 .
• Christian Bär : Elementary Differential Geometry. de Gruyter, Berlin et al. 2001, ISBN 3-11-015519-2 .
• Wolfgang Kühnel : Differential geometry, curves - surfaces - manifolds. 4th, revised edition. Friedr. Vieweg & Sohn, Wiesbaden 2008, ISBN 978-3-8348-0411-2 .

Abstract manifolds, Riemannian geometry

• Rolf Walter: differential geometry. 2nd, revised and expanded edition. BI-Wissenschafts-Verlag, Mannheim et al. 1989, ISBN 3-411-03216-2 .
• Sigurdur Helgason : Differential Geometry, Lie Groups, and Symmetric Spaces (= Graduate Studies in Mathematics. 34). American Mathematical Society, Providence RI, 2001, ISBN 0-8218-2848-7 .
• S. Kobayashi , Katsumi Nomizu: Foundations of Differential Geometry. Volume 1 (= Interscience Tracts in Pure and Applied Mathematics. 15, 1). Interscience Publishers, New York NY et al. 1963.
• Pham Mau Quan: Introduction à la géométrie des variétés différentiables (= Monographies universitaires de mathématiques. 29). Dunod, Paris 1969. ( Content (PDF; 184 kB )).

Differential geometry of the defects

• Hagen Kleinert : Gauge Fields in Condensed Matter. Volume 2: Stresses and Defects. Differential Geometry, Crystal Melting. World Scientific, Singapore et al. 1989, ISBN 9971-5-0210-0 , pp. 743-1456, ( online version ).

Web links

Wikibooks: Differential Geometry  - Learning and Teaching Materials

• Literature on differential geometry in the catalog of the German National Library
• Roberto Torretti:  Nineteenth Century Geometry. In: Edward N. Zalta (Ed.): Stanford Encyclopedia of Philosophy .

Individual evidence

1. ^ M. Atiyah, R. Bott: A Lefschetz fixed point formula for elliptic differential operators. In: Bull. Amer. Math. Soc. 72, 1966, pp. 245-250.
2. ^ R. Palais et al .: Seminar on the Atiyah-Singer index theorem. In: Ann. of Math. Study. No. 57, 1966.
3. ^ S. Smale: What is Global Analysis? In: The American Mathematical Monthly. Vol 76, No. 1, 1969, pp. 4-9.
4. ↑ 58: Global analysis, analysis on manifolds. In: The mathematic atlas. Archived from the original on May 4, 2011 ; accessed on September 4, 2018 .