A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself....

geometry, the unit tangent bundle of a Riemannian manifold (M, g), denoted by T1M, UT(M), UTM, or SM is the unit sphere bundle for the tangent bundle T(M). It...

unit vectors in E x {\displaystyle E_{x}} . When the vector bundle in question is the tangent bundle T M {\displaystyle TM} , the unit sphere bundle is...

the displacement at unit speed along the horocycle tangent to a given unit tangent vector induces a flow on the unit tangent bundle of the hyperbolic plane...

circle bundle. The unit tangent bundle of a non-orientable surface is a circle bundle that is not a principal U ( 1 ) {\displaystyle U(1)} bundle. Only...

V is a unit vector, γ V {\displaystyle \gamma _{V}} remains unit speed throughout, so the geodesic flow is tangent to the unit tangent bundle. Liouville's...

characteristic polynomial of the Jacobi operator of unit tangent vectors is a constant on the unit tangent bundle. It is named after American mathematician Robert...

the unit sphere in the dual number plane. Ball ⁠ n {\displaystyle n} ⁠-sphere Sphere Superellipse Unit circle Unit disk Unit tangent bundle Unit square...

structure on smooth manifolds given by a hyperplane distribution in the tangent bundle satisfying a condition called 'complete non-integrability'. Equivalently...

a line bundle expresses the concept of a line that varies from point to point of a space. For example, a curve in the plane having a tangent line at...

space, PSL(2, R) can be described as the unit tangent bundle of the hyperbolic plane. It is a circle bundle, and has a natural contact structure induced...

setting, a vector field gives a tangent vector at each point of the manifold (that is, a section of the tangent bundle to the manifold). Vector fields...

{\displaystyle p} . Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a global section...

{\displaystyle V_{2}(\mathbb {R} ^{n})} may be identified with the unit tangent bundle to Sn−1. When k = n or n−1 we saw in the previous section that V...

Colin de Verdière states that a compact Riemannian manifold whose unit tangent bundle is ergodic under the geodesic flow is also ergodic in the sense that...

{SO}}(2)} . Alternatively, the bundle of unit-length tangent vectors on the upper half-plane, called the unit tangent bundle, is isomorphic to P S L ( 2...

Kolmogorov. She completed her PhD thesis, titled "Geodesic Flows on Unit Tangent Bundles of Compact Surfaces of Negative Curvature", in 1969. In 1971 she...

O(n)} . The example also works for bundles other than the tangent bundle; if E {\displaystyle E} is any vector bundle of rank k {\displaystyle k} over M...

of its tangent bundle. In particular, a differentiable manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note:...

positive diagonal matrices and N of lower unitriangular matrices on the unit tangent bundle G / Γ. The Ambrose-Kakutani theorem expresses every ergodic flow...

with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold...

T^{1}M} be the tangent bundle of unit-length vectors on the manifold M, and let T 1 H {\displaystyle T^{1}H} be the tangent bundle of unit-length vectors...

the tangent bundle. A choice of affine connection is also equivalent to a notion of parallel transport, which is a method for transporting tangent vectors...

for a given structure group G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M. The notion of G-structures includes various classical...

the set of tangent k-planes in the tangent bundle TM. The target space for the Gauss map N is a Grassmann bundle built on the tangent bundle TM. In the...

J^{2}=-1} when regarded as a vector bundle isomorphism J : T M → T M {\displaystyle J\colon TM\to TM} on the tangent bundle. A manifold equipped with an almost...

defines directional derivative for sections of a vector bundle more general than the tangent bundle. Connections also lead to convenient formulations of...

mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way...

can "solder" a fiber bundle to a smooth manifold, in such a way that the fibers behave as if they were tangent. When the fiber bundle is a homogenous space...

Kolmogorov automorphisms, and the Anosov flow (the geodesic flow on the unit tangent bundle of compact manifolds of negative curvature.) The dyadic map is "shift...