Struik, D. Lectures on Classical Differential Geometry. Weatherburn, C. Differential Geometry of Three Dimensions, 2 vols. Cambridge, England: Cambridge University Press, Weisstein, E. Weisstein, Eric W. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. Walk through homework problems step-by-step from beginning to end. Hints help you try the next step on your own.
The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.
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Many concepts and techniques of analysis and differential equations have been generalized to the setting of Riemannian manifolds. A distance-preserving diffeomorphism between Riemannian manifolds is called an isometry. This notion can also be defined locally , i.
Any two regular curves are locally isometric. However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same. In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat.
An important class of Riemannian manifolds is the Riemannian symmetric spaces , whose curvature is not necessarily constant. These are the closest analogues to the "ordinary" plane and space considered in Euclidean and non-Euclidean geometry. Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite.
A special case of this is a Lorentzian manifold , which is the mathematical basis of Einstein's general relativity theory of gravity. Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric , that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds. Symplectic geometry is the study of symplectic manifolds. An almost symplectic manifold is a differentiable manifold equipped with a smoothly varying non-degenerate skew-symmetric bilinear form on each tangent space, i.
A diffeomorphism between two symplectic manifolds which preserves the symplectic form is called a symplectomorphism. Non-degenerate skew-symmetric bilinear forms can only exist on even-dimensional vector spaces, so symplectic manifolds necessarily have even dimension. In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism. The phase space of a mechanical system is a symplectic manifold and they made an implicit appearance already in the work of Joseph Louis Lagrange on analytical mechanics and later in Carl Gustav Jacobi 's and William Rowan Hamilton 's formulations of classical mechanics.
By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic. The only invariants of a symplectic manifold are global in nature and topological aspects play a prominent role in symplectic geometry. Birkhoff in It claims that if an area preserving map of an annulus twists each boundary component in opposite directions, then the map has at least two fixed points.
Contact geometry deals with certain manifolds of odd dimension.
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It is close to symplectic geometry and like the latter, it originated in questions of classical mechanics. A local 1-form on M is a contact form if the restriction of its exterior derivative to H is a non-degenerate two-form and thus induces a symplectic structure on H p at each point. A contact analogue of the Darboux theorem holds: all contact structures on an odd-dimensional manifold are locally isomorphic and can be brought to a certain local normal form by a suitable choice of the coordinate system.
Complex differential geometry is the study of complex manifolds. An almost complex manifold is complex if and only if it admits a holomorphic coordinate atlas. An almost Hermitian structure is given by an almost complex structure J , along with a Riemannian metric g , satisfying the compatibility condition.
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An almost Hermitian structure defines naturally a differential two-form. CR geometry is the study of the intrinsic geometry of boundaries of domains in complex manifolds. Differential topology is the study of global geometric invariants without a metric or symplectic form. Differential topology starts from the natural operations such as Lie derivative of natural vector bundles and de Rham differential of forms.
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Beside Lie algebroids , also Courant algebroids start playing a more important role. A Lie group is a group in the category of smooth manifolds. Beside the algebraic properties this enjoys also differential geometric properties. The most obvious construction is that of a Lie algebra which is the tangent space at the unit endowed with the Lie bracket between left-invariant vector fields.
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Beside the structure theory there is also the wide field of representation theory. The apparatus of vector bundles , principal bundles , and connections on bundles plays an extraordinarily important role in modern differential geometry. A smooth manifold always carries a natural vector bundle, the tangent bundle. Loosely speaking, this structure by itself is sufficient only for developing analysis on the manifold, while doing geometry requires, in addition, some way to relate the tangent spaces at different points, i. An important example is provided by affine connections.
For a surface in R 3 , tangent planes at different points can be identified using a natural path-wise parallelism induced by the ambient Euclidean space, which has a well-known standard definition of metric and parallelism. In Riemannian geometry , the Levi-Civita connection serves a similar purpose. The Levi-Civita connection defines path-wise parallelism in terms of a given arbitrary Riemannian metric on a manifold.
More generally, differential geometers consider spaces with a vector bundle and an arbitrary affine connection which is not defined in terms of a metric. In physics, the manifold may be the space-time continuum and the bundles and connections are related to various physical fields. From the beginning and through the middle of the 18th century, differential geometry was studied from the extrinsic point of view: curves and surfaces were considered as lying in a Euclidean space of higher dimension for example a surface in an ambient space of three dimensions.
The simplest results are those in the differential geometry of curves and differential geometry of surfaces. Starting with the work of Riemann , the intrinsic point of view was developed, in which one cannot speak of moving "outside" the geometric object because it is considered to be given in a free-standing way. The fundamental result here is Gauss's theorema egregium , to the effect that Gaussian curvature is an intrinsic invariant. The intrinsic point of view is more flexible. For example, it is useful in relativity where space-time cannot naturally be taken as extrinsic what would be "outside" of it?
However, there is a price to pay in technical complexity: the intrinsic definitions of curvature and connections become much less visually intuitive. These two points of view can be reconciled, i. See the Nash embedding theorem. In the formalism of geometric calculus both extrinsic and intrinsic geometry of a manifold can be characterized by a single bivector-valued one-form called the shape operator.