In algebraic geometry, in contrast, there is an intrinsic definition of tangent space at a point P of a variety V, that gives a vector space of dimension at least that of V. The points P at which the dimension is exactly that of V are called the non-singular points; the others are singular points. For example, a curve that crosses itself doesn't have a unique tangent line at that point. The singular points of V are those where the 'test to be a manifold' fails.
Once tangent spaces have been introduced, one can define vector fields, which are abstractions of the velocity field of particles moving on a manifold. A vector field attaches to every point of the manifold a vector from the tangent space at that point, in a smooth manner. Such a vector field serves to define a generalized ordinary differential equation on a manifold: a solution to such a differential equation is a differentiable curve on the manifold whose derivative at any point is equal to the tangent vector attached to that point by the vector field.
All the tangent spaces can be "glued together" to form a new differentiable manifold of twice the dimension, the tangent bundle of the manifold.
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1.1 Definition as directions of curves
2 Properties1.2 Definition via derivations 1.3 Definition via the cotangent space |
There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straight forward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.
Suppose M is a Ck manifold (k ≥ 1) and p is a point in M. Pick a chart φ : U → Rn where U is an open subset of M containing p. Suppose two curves γ1 : (-1,1) → M and γ2 : (-1,1) → M with γ1(0) = γ2(0) = p are given such that φ o γ1 and φ o γ2 are both differentiable at 0. Then γ1 and γ2 are called tangent at 0 if the ordinary derivatives of φ o γ1 and φ o γ2 at 0 coincide. This is an equivalence relation, and the equivalence classes are known as the tangent vectors of M at p. The equivalence class of the curve γ is written as γ'(0). The tangent space of M at p, denoted by TpM, is defined as the set of all tangent vectors; it does not depend on the choice of chart φ.
To define the vector space operations on TpM, we use a chart φ : U → Rn and define the map (dφ)p : TpM → Rn by (dφ)p(γ'(0)) = (φ o γ)'(0). It turns out that this map is bijective and can thus be used to transfer the vector space operations from Rn over to TpM, turning the latter into an n-dimensional real vector space. Again, one needs to check that this construction does not depend on the particular chart φ chosen, and in fact it does not.
Supppose M is a C∞ manifold. A real-valued function g : M → R belongs to C∞(M) if g o φ-1 is infinitely often differentiable for every chart φ : U → Rn. C∞(M) is a real associative algebra.
Pick a point p in M. A derivation at p is a linear map D : C∞(M) → R which has the property that for all g, h in C∞(M):
The relation between the tangent vectors defined earlier and derivations is as follows: if γ is a curve with tangent vector γ'(0), then the corresponding derivation is D(g) = (g o γ)'(0) (where the derivative is taken in the ordinary sense, since g o γ is a function from (-1,1) to Rn).
Again we start with a C∞ manifold M and a point p in M. Consider the ideal I in C∞(M) consisting of all functions g such that g(p) = 0. Then I and I 2 are real vector spaces, and TpM may be defined as the dual space of the quotient space I / I 2. This latter quotient space is also known as the cotangent space of M at p.
While this definition is the most abstract, it is also the one most easily transferred to other settings, for instance to the varieties considered in algebraic geometry.
If D is a derivation, then D(g) = 0 for every g in I2, and this means that D gives rise to a linear map I / I2 → R. Conversely, if r : I / I2 → R is a linear map, then D(g) = r((g - g(p)) + I 2) is a derivation. This yields the correspondence between the tangent space defined via derivations and the tangent space defined via the cotangent space.
If M is an open subset of Rn, then M is a C∞ manifold in a natural manner (take the charts to be the identity maps), and the tangent spaces are all naturally identified with Rn.
Every differentiable map f : M → N between Ck manifolds induces natural linear maps between the corresponding tangent spaces:
If g : M → R is an element of C∞(M) and v is a tangent vector of M at p, then we can define the directional derivative of g at p in the direction v. The result is a number, written as Dv(g). If we think of v as the direction of a curve, v = γ'(0), then
Formal definitions
Definition as directions of curves
Definition via derivations
modeled on the product rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space TpM. Definition via the cotangent space
Properties
defined by
if the tangent space is defined via curves and by
if the tangent space is defined via derivations.
The linear map (df)p is called the differential of f at p; in a sense it is the best linear approximation to f near p.
and if v is thought of as a derivation v = D, then