In mathematics, the tangent space of a manifold is a generalization of tangent lines to curves in two-dimensional space and tangent planes to surfaces...

cotangent bundle of the manifold. The tangent space and the cotangent space at a point are both real vector spaces of the same dimension and therefore isomorphic...

A similar definition applies to space curves and curves in n-dimensional Euclidean space. The point where the tangent line and the curve meet or intersect...

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...

In algebraic geometry, the Zariski tangent space is a construction that defines a tangent space at a point P on an algebraic variety V (and more generally)...

tangent space at each event is a vector space of the same dimension as spacetime, 4. In practice, one need not be concerned with the tangent spaces....

metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together...

is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that φ : M → N {\displaystyle \varphi \colon M\to N} is a smooth...

reuse is made possible by encoding maps in tangent space. The tangent space is a vector space, which is tangent to the model's surface. The coordinate system...

a tangent frame), is an ordered basis of particular tangent space. Likewise, a tangent frame is a linear isomorphism of Rn to this tangent space. A moving...

of curves in Rn. More generally, tangent vectors are elements of a tangent space of a differentiable manifold. Tangent vectors can also be described in...

In geometry, the tangent cone is a generalization of the notion of the tangent space to a manifold to the case of certain spaces with singularities. In...

of M fixing p and acting on the tangent space T p M {\displaystyle T_{p}M} as minus the identity (every symmetric space is complete, since any geodesic...

vector space over that field (the cotangent space). The tangent space T x ( X ) {\displaystyle T_{x}(X)} is defined as the dual of this vector space. The...

Euclidean metric on the tangent spaces at the points of the manifold (these tangent spaces are thus Euclidean vector spaces). This results in a Riemannian...

positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity...

algebraic geometry, the tangent space to a functor generalizes the classical construction of a tangent space such as the Zariski tangent space. The construction...

space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space....

about the underlying topological space. For example, the tangent bundle consists of the collection of tangent spaces parametrized by the points of a differentiable...

geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions...

an injective function at every point p of M (where TpX denotes the tangent space of a manifold X at a point p in X and Dp f is the derivative (pushforward)...

transformation on each tangent space, then the manifold M is orientable. Conversely, M is orientable if and only if the structure group of the tangent bundle can...

functional on the tangent space at each point. A Riemannian manifold, or Riemann space, is a smooth manifold whose tangent spaces are endowed with inner...

manifold M, and a vector in the tangent space to M at a point of N, it can be decomposed into the component tangent to N and the component normal to...

geometric measure theory an approximate tangent space is a measure theoretic generalization of the concept of a tangent space for a differentiable manifold. In...

vectors from one tangent space to another. Parallel transport of tangent vectors is a way of moving vectors from one tangent space to another along a...

that the tangent space T Q {\displaystyle TQ} corresponds to the velocities of the points q ∈ Q {\displaystyle q\in Q} , while the cotangent space T ∗ Q...

The fibre of the holomorphic tangent bundle over a point is the holomorphic tangent space, which is the tangent space of the underlying smooth manifold...

to "slide along" that submanifold. They span the tangent space of that submanifold. The tangent space T p N {\displaystyle T_{p}N} should have the same...

a (possibly asymmetric) Minkowski norm F(x, −) is provided on each tangent space TxM, that enables one to define the length of any smooth curve γ : [a...