- Tangent space
In

mathematics , the**tangent space**of amanifold is a concept which facilitates the generalization of vectors fromaffine space s to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other.**Informal description**In

differential geometry , one can attach to every point "x" of a differentiablemanifold a**tangent space**, a realvector space which intuitively contains the possible "directions" in which one can pass through "x". The elements of the tangent space are called**tangent vectors**at "x". All the tangent spaces have the same dimension, equal to the dimension of the manifold.For example, if the given manifold is a 2-

sphere , one can picture the tangent space at a point as the plane which touches the sphere at that point and isperpendicular to the sphere's radius through the point. More generally, if a given manifold is thought of as an embedded submanifold ofEuclidean space one can picture the tangent space in this literal fashion.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. SeeZariski tangent space .Once tangent spaces have been introduced, one can define

vector field s, 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 generalizedordinary differential equation on a manifold: a solution to such a differential equation is a differentiablecurve 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.**Formal definitions**There are various equivalent ways of defining the tangent spaces of a manifold. While the definition via directions of curves is quite straightforward given the above intuition, it is also the most cumbersome to work with. More elegant and abstract approaches are described below.

**Definition as directions of curves**Suppose "M" is a C

^{"k"}manifold ("k" ≥ 1) and "x" is a point in "M". Pick a chart φ : "U" →**R**^{"n"}where "U" is an open subset of "M" containing "x". Suppose two curves γ_{1}: (-1,1) → "M" and γ_{2}: (-1,1) → "M" with γ_{1}(0) = γ_{2}(0) = "x" are given such that φ o γ_{1}and φ o γ_{2}are both differentiable at 0. Then γ_{1}and γ_{2}are called "tangent at 0" if the ordinary derivatives of φ o γ_{1}and φ o γ_{2}at 0 coincide. This defines anequivalence relation on such curves, and theequivalence class es are known as the tangent vectors of "M" at "x". The equivalence class of the curve γ is written as γ'(0). The tangent space of "M" at "x", denoted by T_{"x"}"M", 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 T

_{"x"}"M", we use a chart φ : "U" →**R**^{"n"}and define the map (dφ)_{"x"}: T_{"x"}"M" →**R**^{"n"}by (dφ)_{"x"}(γ'(0)) = $scriptstylefrac\{d\}\{dt\}$(φ o γ)(0). It turns out that this map isbijective and can thus be used to transfer the vector space operations from**R**^{"n"}over to T_{"x"}"M", 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.**Definition via derivations**Suppose "M" is a C

^{∞}manifold. A real-valued function "f" : "M" →**R**belongs to C^{∞}("M") if "f" o φ^{-1}is infinitely often differentiable for every chart φ : "U" →**R**^{"n"}. C^{∞}("M") is a realassociative algebra for thepointwise product and sum of functions and scalar multiplication.Pick a point "x" in "M". A "derivation" at "x" is a

linear map "D" : C^{∞}("M") →**R**which has the property that for all "f", "g" in C^{∞}("M")::"D"("fg") = "D"("f")·"g"("x") + "f"("x")·"D"("g")modeled on theproduct rule of calculus. These derivations form a real vector space in a natural manner; this is the tangent space T_{"x"}"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"("f") = ("f" o γ)'(0) (where the derivative is taken in the ordinary sense, since "f" o γ is a function from (-1,1) to

**R**).Generalizations of this definition are possible, for instance to

complex manifold s and algebraic varieties. However, instead of examining derivations "D" from the full algebra of functions, one must instead work at the level of germs of functions. The reason is that thestructure sheaf may not be fine for such structures. For instance, let "X" be an algebraic variety withstructure sheaf "F". Then theZariski tangent space at a point "p"∈"X" is the collection of "K"-derivations "D":"F"_{p}→"K", where "K" is the groundfield and "F"_{p}is the stalk of "F" at "p".**Definition via the cotangent space**Again we start with a C

^{∞}manifold "M" and a point "x" in "M". Consider the ideal "I" in C^{∞}("M") consisting of all functions "f" such that "f"("x") = 0. Then "I" and "I"^{ 2}are real vector spaces, and T_{"x"}"M" may be defined as thedual space of the quotient space "I" / "I"^{ 2}. This latter quotient space is also known as thecotangent space of "M" at "x".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"("f") = 0 for every "f" in "I"

^{2}, and this means that "D" gives rise to a linear map "I" / "I"^{2}→**R**. Conversely, if "r" : "I" / "I"^{2}→**R**is a linear map, then "D"("f") = "r"(("f" - "f"("x")) + "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.**Properties**If "M" is an open subset of

**R**^{"n"}, then "M" is a C^{∞}manifold in a natural manner (take the charts to be theidentity map s), and the tangent spaces are all naturally identified with**R**^{"n"}.**Tangent vectors as directional derivatives**One way to think about tangent vectors is as

directional derivative s. Given a vector "v" in**R**^{"n"}one defines the directional derivative of a smooth map "f" :**R**^{"n"}→**R**at a point "x" by:$scriptstyle\; D\_v\; f(x)\; =\; frac\{d\}\{dt\}igg|\_\{t=0\}f(x+tv)=sum\_\{i=1\}^\{n\}v^ifrac\{partial\; f\}\{partial\; x^i\}(x).$This map is naturally a derivation. Moreover, it turns out that every derivation of C^{∞}(**R**^{"n"}) is of this form. So there is a one-to-one map between vectors (thought of as tangent vectors at a point) and derivations.Since tangent vectors to a general manifold can be defined as derivations it is natural to think of them as directional derivatives. Specifically, if "v" is a tangent vector of "M" at a point "x" (thought of as a derivation) then define the directional derivative in the direction "v" by:$scriptstyle\; D\_v(f)\; =\; v(f),$where "f" : "M" →

**R**is an element of C^{∞}("M").If we think of "v" as the direction of a curve, "v" = γ'(0), then we write:$scriptstyle\; D\_v(f)\; =\; (fcircgamma)\text{'}(0).$**The derivative of a map**"Main article:

Pushforward (differential) "Every smooth (or differentiable) map "φ" : "M" → "N" between smooth (or differentiable) manifolds induces natural

linear map s between the corresponding tangent spaces::$scriptstyle\; mathrm\; dvarphi\_xcolon\; T\_xM\; o\; T\_\{varphi(x)\}N.$If the tangent space is defined via curves, the map is defined as:$scriptstylemathrm\; dvarphi\_x(gamma\text{'}(0))\; =\; (varphicircgamma)\text{'}(0).$If instead the tangent space is defined via derivations, then:$scriptstylemathrm\; dvarphi\_x(X)(f)\; =\; X(fcirc\; varphi).$The linear map d"φ"

_{"x"}is called variously the "derivative", "total derivative", "differential", or "pushforward" of "φ" at "x". It is frequently expressed using a variety of other notations::$scriptstyle\; Dvarphi\_x,quad\; (varphi\_*)\_x,quad\; varphi\text{'}(x).$In a sense, the derivative is the best linear approximation to "φ" near "x". Note that when "N" =**R**, the map d"φ"_{"x"}: T_{"x"}"M"→**R**coincides with the usual notion of the differential of the function "φ". Inlocal coordinates the derivative of "f" is given by theJacobian .An important result regarding the derivative map is the following::

**Theorem**. If "φ" : "M" → "N" is alocal diffeomorphism at "x" in "M" then d"φ"_{"x"}: T_{"x"}"M" → T_{"φ"("x")}"N" is a linearisomorphism . Conversely, if d"φ"_{"x"}is an isomorphism then there is an open neighborhood "U" of "x" such that "φ" maps "U" diffeomorphically onto its image.This is a generalization of theinverse function theorem to maps between manifolds.**References*** ("to appear").

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