Tangent Space & Vector Fields

Tangent Space

For a general manifold, not necessary embedded in R^m, we define the notion of tangent space by abstracting the notion of the directional derivative. The tangent space at m to M , denoted T_m(M ), is the set of all linear maps X from C^∞(M ) into R satisfying:

• "Product rule": X(fg) = X(f)g(m) + f(m)X(g) for all f and g in C^∞(M );


• "Localization" : if f is equal to g in a neighborhood of m, then X(f) = X(g).

This is a real vector space. An element of T_m(M ) is called a tangent vector at m.

If x₁, ..., x_n is a local coordinate system, then each tangent vector X at m is expressed uniquely as

X(f)  = ∑ a_k ∂f / ∂x_k(m), for k = 1, n and a₁, ..., a_n real constants. This means T_m(M ) is a real vector space of dimension n.

Vector Fields

A vector field is a map X that associates to each point m in M a tangent vector X_m ∈ T_m(M ). A vector field can be expressed 

X_m(f)  = ∑ a_k(m) ∂f / ∂x_k, k = 1, n

where a_k's are real-valued functions. A vector field is smooth if the coefficient functions a_k are smooth in each local coordinate system. 

If we apply a vector field to a smooth function f by applying X_m to f at each point m. The result X(f) is then another function, which will be smooth if X is a smooth vector field. A smooth vector field is a map from  C^∞(M ) -->  C^∞(M ) that satisfies the product rule : X(fg) = fX(g) + X(f)g. Here X(fg) is a function, not a number, and that on the right-hand side, we do not evaluate f or g at any point. The equation X(fg) can be restated as saying that a vector field is a derivation of the algebra of smooth functions.

Therefore, we can think of the vector field as a first-order differential operator (mapping the space of smooth functions to itself), the one obtained by differentiating a function at each point in the direction of the tangent vector at that point (we can also see a geometric picture of a vector field as a collection of arrows, one at each point in the manifold). If we multiply two vector fields, we will get a second-order differential operator; that is not a vector field. However, if X and Y are vector fields and we compute their commutator XY - YX, then the second-order items in XY will cancel with the second-order terms in YX and the result will again be first-order differential operator. The space of smooth vector fields then becomes an infinite-dimensional Lie algebra with the bracket defined by [X, Y] = XY - YX. 

If X is a vector field and ϒ : (a,b) --> M  is a smooth curve in M , then ϒ is called an integral curve for X if for each t ∈ (a,b), we have dϒ/dt = X_ϒ(t). In a smooth local coordinate system x₁, ..., x_n, ϒ(t) will be represented a family of functions x₁(t), ..., x_n(t) and the vector field X will be represented in the form of X_m(f) as above, with each a_k being a smooth function of x₁, ..., x_n.  Here, for all smooth functions f on M,

dϒ/dt (f) = df(ϒ(t))/dt. By chain rule,  df(ϒ(t))/d = ∑ ∂f / ∂x_k dx_k/dt.

Compare to X_m(f)  = ∑ a_k(m) ∂f / ∂x_k, k = 1, n, we have 

dx_k(t) / dt = a_k(x₁(t), ..., x_n(t)). This is a equation of first-order ordinary differential equations.

Flow Along a Vector Field

A vector field X is called complete if ϒ(t) can be defined for all t for all initial points m. If X is a complete vector field, then one can define the associated flow on M. This is a family of maps Φ_t: M  --> M  defined so that if ϒ is an integral curve for X which ϒ(0) = m, then Φ_t(m) =  ϒ(t). This means that Φ_t(m) is defined by starting at m and "flowing" along the vector field X for time t. (If X is not complete, one can still define a sort of flow, but then each Φ_t is defined only on part of M.) If X is a smooth complete vector field, then each Φ_t is a smooth map of M to itself, and the maps satisfy Φ_t o Φ_s = Φ_t+s.

Submanifolds of Vector Fields

Suppose V is a real vector space of dimension n. A subset M  of V is called a smooth embedded submanifold of dimension k if given any m₀ in M  , there exists a smooth coordinate system (Φ, U) defined in the neighborhood U of  m₀ such that for any m ∈ U, m is in U ∩ M if and only if  Φ(m) is in R^k ⊂ Rⁿ. Here, R^k is the subset if Rⁿ where the last n - k coordinates are zero. This says that locally, in a suitable coordinate system, M  looks like R^k sitting inside Rⁿ. If M  is a smooth embedded submanifold of dimension k, then we can make M  into a smooth manifold of dimension k as follows. 

If M  is a smooth embedded submanifold of V, then the inclusion map i of M  into V is a smooth map. The differential i_* : T_m(M ) --> T_m(V) is injective, and it is customary to identify T_m(M ) with its image  T_m(V ), which is a k-dimensional subspace of the n-dimensional space T_m(V ). This allows us to think of the tangent space to M  at m as a subspace of the tangent space to V at m. However, we are identifying the tangent space at m to V with V itself. Thus, the tangent space to M  at m is identified with a subspace of V.