Dr. Wang Xingbo Fall ， 2005 Mathematical & Mechanical Method in Mechanical Engineering

1.Review of Topology 2.Concepts of manifolds Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds

How can we describe it? Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds

Many engineering objects have a shape of complicated surface. These complicated surfaces can be described by manifolds. Theories of manifolds have exhibited their elegances and excellences in many aspects of engineering, e.g. in controlling of robots, in structural analysis of mechanical engineering. Manifold is regarded to be a powerful tool for a senior engineer or a researcher to master.Many engineering objects have a shape of complicated surface. These complicated surfaces can be described by manifolds. Theories of manifolds have exhibited their elegances and excellences in many aspects of engineering, e.g. in controlling of robots, in structural analysis of mechanical engineering. Manifold is regarded to be a powerful tool for a senior engineer or a researcher to master. Mathematical & Mechanical Method in Mechanical Engineering An Introduction to Manifolds

A topological space is a pair (X, T) where X is a set and T is a class of subsets of X, called topology, which satisfies the following three properties. (i) X,  ∈ T. (ii) If { X i } i ∈ I ∈ T, then ∪ i ∈ I X i ∈ T (iii) If X 1, …, X n ∈ T, then ∩ i=1, …,n X i ∈ T. Mathematical & Mechanical Method in Mechanical Engineering Review of Topology

If (X, T) is a topological space, the elements of T are said open sets. A subset K of X is said closed if its complementary set X \K is open A subset K of X is said closed if its complementary set X \K is open The closure of a set U X is the intersection of all the closed sets K X with U K Mathematical & Mechanical Method in Mechanical Engineering Review of Topology

Mathematical & Mechanical Method in Mechanical Engineering Closure of a set

If (X, T ) and (Y, U ) are topological spaces, a mapping f : X → Y is said continuous if is open for each T ∈ U Mathematical & Mechanical Method in Mechanical Engineering Continuous function

An injective, surjective and continuous mapping f : X→Y, whose inverse mapping is also continuous, is said homeomorphism from X to Y. If there is a homeomorphism from X to Y these topological spaces are said homeomorphic. Mathematical & Mechanical Method in Mechanical Engineering Homeomorphism

If (X,T) is a topological space, a class B T is said base of the topology, if each open set turns out to be union of elements of B. A topological space which admits a countable base of its topology is said second countable. If (X,T) is second countable, from any base B it is possible to extract a subbase B ’ B which is countable. Mathematical & Mechanical Method in Mechanical Engineering Base, second countable

If A is a class of subsets of X≠  ; and C A is the class of topologies T on X with A  T, T A :=  T ∈ CA T is said the topology generated by A. Notice that C A ≠  because the set of parts of X, P(X), is a topology and includes A. Mathematical & Mechanical Method in Mechanical Engineering Topology generated by set-class

If A  X, where (X, T ) is a topological space, the pair (A, T A ) where, T A := {U  A | U  T },defines a topology on A which is said the topology induced on A by X. Mathematical & Mechanical Method in Mechanical Engineering topology induced on a set

If (X,T) is a topological space and p  X, a neighborhood of p is an open set U  X with p ∈ U. If X and Y are topological spaces and x  X, f: X→Y is said to be continuous in X, if for every neighborhood of f(x), V  Y, there is a neighborhood of x, U  X, such that f(U)  V. It is simply proven that f : X→Y as above is continuous if and only if it is continuous in every point of X. Mathematical & mechanical Method in Mechanical Engineering Neighborhood

A topological space (X,T) is said connected if there are no open sets A, B≠  with A  B =  and A  B = X. It turns out that if f: X→Y is continuous and the topological space X is connected, then f(Y) is a connected topological space when equipped with the topology induced by the topological space Y. Mathematical & mechanical Method in Mechanical Engineering Connect

A topological space (X, T ) is said Hausdorff if each pair (p,q)  X  X admits a pair of neighborhoods U p, U q with p ∈ U p, q ∈ U q and U p  U q = . If X is Hausdorff and x  X is a limit of the sequence {X n } n ∈ N  X, this limit is unique. Mathematical & mechanical Method in Mechanical Engineering Hausdorff

A semi metric space is a set X endowed with a semidistance. d: X  X → [0,+ ∞], with Mathematical & mechanical Method in Mechanical Engineering Semi-distance

The semidistance is called distance and the semi metric space is called metric space. An open metric balls are defined as Mathematical & mechanical Method in Mechanical Engineering Open Ball

A topological space (X,T) is said connected by paths if, for each pair p, q  X there is a continuous path  : [0,1] →X such that  (0) = p,  (1) = q, Mathematical & mechanical Method in Mechanical Engineering Connected by path

Mathematical & mechanical Method in Mechanical Engineering Cover If X is any set, a covering of X is a class {X i } i ∈ I, X i  X for all i  I, such that  i ∈ I X i = X

Mathematical & mechanical Method in Mechanical Engineering Compactness-Finite Cover A topological space (X, T ) is said compact if from each covering of X, {X i } i ∈ I are made of open sets, it is possible to extract a covering {X j } j ∈ I of X with j finite. This is also called a finite covering property

Mathematical & mechanical Method in Mechanical Engineering Group Let G be a set and  be a operation defined on W. If W and  satisfy the following regulations: 1.There is a unit e in G such that where Then G is called a group over R

Mathematical & mechanical Method in Mechanical Engineering isomorphism Let S and T be tow groups with operations ,  respectively. If there exists a one-to-one mapping  : S  T such that, for any 1. If it results in 2. If are unit in S and T respectively, then then S is said to be isomorphic to T, or vice versa; the mapping  is said to be a isomorphism between S and T. Two isomorphic groups can be regarded to have the same structure algebraically

Mathematical & mechanical Method in Mechanical Engineering Concepts of Manifolds A topological space (X, T ) is said topological manifold of dimension n if X is Hausdorff, second countable and is locally homeomorphic to R n, i.e., for every p  X there is a neighborhood p  U p and a homomorphism  p : U p → V p where V p  R n is a open set.

(n-chart)Let X be topological space, U  is an open subset of X. Let   be a homeomorphism from U   X to an open subset V   R n, namely,   : p→(x 1 (p), …,x n (p)). Then the ordered pair (U ,   )= C  is called an n-chart on M. where R n is the n-dimensional Euclidean space. A chart can be thought of a mapping from some open set to an open subset of R n Mathematical & mechanical Method in Mechanical Engineering Chart

Mathematical & mechanical Method in Mechanical Engineering Chart

Let (U ,   ) and (U ,   ) be two charts on a topological space M. If U   U  , let V  and V  be image of U   U  under corresponding homeomorphisms   and  . The two charts are said to be compatible if     -1 viewed as a mapping from V   R n to V   R n, is a C  function. If U   U  =  then the charts are also said to be compatible. If     -1 and     -1 are all C k (k<  ) functions, then   and   are said to C k - compatible. If any   and   are said to C  - compatible, then M is said to be smooth. Mathematical & mechanical Method in Mechanical Engineering K-Compatible

Mathematical & mechanical Method in Mechanical Engineering k-Compatible

An atlas A on a topological space M is a collection of charts{C  } on M such that 1. Any two charts in atlas are piecewise k- compatible; 2.A covers M, i.e. Mathematical & Mechanical Method in Mechanical Engineering Atlas Atlas

A differential structure on a topological space is an atlas with the property that any chart that is compatible with the charts of the atlas is also an element of the atlas. Mathematical & Mechanical Method in Mechanical Engineering Differential structure, Differentian Manifolds Differential structure, Differentian Manifolds An n-dimensional differential manifold M is a topological space endowed with a differential structure of n-charts.

If M is a n-dimensional differential manifold, then any point P  M has such a open neighborhood U that is homeomorphism to an open set V of R n, or we can say, that there exists at least one open subset U of M that has a n-chart (U,  ) such that  (P)=V  R n. At this time, the coordinate  ((P)) i of image  (P) corresponding to P is called coordinate of P  U and is denoted by x i (P)=(  (P)) i. (U, x i )is called a local coordinate system. Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems Local Cooridnate Systems

Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems Local Cooridnate Systems It can be seen that, two charts (U,  ), (V,  ) on an n- dimensional differential manifold M are related two local coordinate systems. If U  V  ， then there also exist two local coordinate systems corresponding to U  V. Thus any point P  U  V has two coordinate representations x i (P)=(  (P)) i and y i (P)=(  (P)) i and the two are dependent.

Mathematical & Mechanical Method in Mechanical Engineering Local Cooridnate Systems Local Cooridnate Systems

Mathematical & Mechanical Method in Mechanical Engineering Differentiable Partitions of Unity on Manifolds Differentiable Partitions of Unity on Manifolds

Tangent, Tangent Bundle and State Space. Mathematical & Mechanical Method in Mechanical Engineering Tensor Fields in Manifolds and Associated Geometric Structures Tensor Fields in Manifolds and Associated Geometric Structures Local representative Let be a continuous function from R to a differential manifold M.

.. Local representative Mathematical & Mechanical Method in Mechanical Engineering Tangent, Tangent Bundle and State Space Tangent, Tangent Bundle and State Space

Two curves f and g are said to be related at p if and only if 1. f(0)=g(0)=p; 1. f(0)=g(0)=p; 2.The derivatives of the local representations of f and g are equal 2.The derivatives of the local representations of f and g are equal Mathematical & Mechanical Method in Mechanical Engineering Related

If f(t) and g(t) are related in chart (U ,   ), they are also related in chart (V ,   ) Mathematical & Mechanical Method in Mechanical Engineering Related properties Related properties

Mathematical & Mechanical Method in Mechanical Engineering Related properties Related properties f(t) and g(t) are related in chart (U ,   )

Mathematical & Mechanical Method in Mechanical Engineering Related properties

Mathematical & Mechanical Method in Mechanical Engineering Related Properties Related Properties

If M is a differentiable manifold and p  M, the tangent space at point p, denoted as T p M, is defined to be the set of all equivalent classes Q p at p in M. Mathematical & Mechanical Method in Mechanical Engineering Tangent space Tangent space T p M has the same dimension as M Define a map is injective

For any v in R n, choose  such that for any |t|<  Mathematical & Mechanical Method in Mechanical Engineering Tangent space Tangent space is a path through in and is a smooth path through p is bijective, a linear isomorphic map from T p M to R n

Let M be a differentiable manifold, p  M, and take a chart (U,  ) with p  U. If E 1,…,E n is the canonical basis of R n, then define a basis in T p M which we call the basis induced in T p M by the chart (U,  ) Mathematical & Mechanical Method in Mechanical Engineering Basis induced by a chart (U,   ), ( V,   ) with p  U,V and induced basis on T p M

Let M be a differentiable manifold. A derivation in T p M is a R-linear map D p : D(M) → R, such that, for each pair f, g  D(M): Mathematical & Mechanical Method in Mechanical Engineering Derivations Derivations Symbol D(M) indicates the real vector space of all differential functions from manifold M to R indicate the vector space spanned by Symbol D p M is used to indicate the R-vector space of the derivations in p

Let M be a differential manifold. Take any T p M and any D p  D p M Let M be a differential manifold. Take any T p M and any D p  D p M (1) If h  D(M) vanishes in a open neighborhood of p or, more strongly, h = 0 in the whole manifold M,then D p h= 0 (2) For every f, g  D(M), D p f = D p g provide f(q) = g(q) in an open neighborhood of p. Mathematical & Mechanical Method in Mechanical Engineering Derivation Derivation

If f: B→R is C ∞ (B) where B  R n is an open starshaped neighborhood of If f: B→R is C ∞ (B) where B  R n is an open starshaped neighborhood of, then there are n differentiable mappings g i : B→R such that, if, then, then there are n differentiable mappings g i : B→R such that, if, then Mathematical & Mechanical Method in Mechanical Engineering Flander's Lemma Flander's Lemma

Mathematical & Mechanical Method in Mechanical Engineering Flander’s lemma Flander’s lemma

Mathematical & Mechanical Method in Mechanical Engineering basis of T p M basis of T p M Let M be a differentiable manifold and p  M. There exists a R- value vector space isomorphism F: T p M  D p M such that, if is the basis of T p M induced by any local coordinate system about p with coordinates (x 1,..., x n ), it holds: And in particular the set is a basis of D p M

The tangent bundle of a manifold M, denoted by TM is defined as the union of the tangent spaces for all p  M. That is: The tangent bundle of a manifold M, denoted by TM is defined as the union of the tangent spaces for all p  M. That is: Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle Tangent Bundle

TM is itself a differential manifold of dimension 2n TM is itself a differential manifold of dimension 2n TM= {(p, v) |p  M, v  T p M} Tangent bundle is called a state space Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle Tangent Bundle

Given two manifolds A and B and a function f:A  B, Given two manifolds A and B and a function f:A  B, there is a natural way to form a mapping, denoted by Tf, from TA to TB Mathematical & Mechanical Method in Mechanical Engineering Tangent Bundle Tangent Bundle

Let M be an n-dimensional manifold. For each Let M be an n-dimensional manifold. For each p  M, the dual space is called the cotangent space on p and its elements are called co- tangent vectors or differential 1-forms on p. If (x 1,..., x n ) are coordinates about p inducing the basis, the associated dual basis in is denoted by {dx k | p } k=1, …,n. Mathematical & Mechanical Method in Mechanical Engineering Cotangent Space and Phase Space Cotangent Space and Phase Space

Mathematical & Mechanical Method in Mechanical Engineering Cotangent Space and Phase Space Cotangent Space and Phase Space The cotangent bundle of a manifold M, denoted by T * M is defined as the union of the cotangent spaces for all p  M. That is: A cotangent space is also called a phase space that is a collection of all possible positions and momenta that cab be obtained by a configuration space.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection Let M be a differentiable manifold. An affine connection or covariant derivative r, is a map where X, Y,  Y X are differentiable contravariant vector fields on M, which obeys the following requirements: (1)  fY +gZ X = f ▽ Y X + g  Z X, for all differentiable functions f, g and differentiable vector fields X, Y,Z ; (2)  Y fX = Y(f)X+f  Y X for all differentiable vector field X, Y and differentiable functions f, (3)  X (  Y +  Z) =  X Y +  X Z for all ,  R and differentiable vector fields X, Y, Z.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection In components referred to any local coordinate system if i, j are fixed define a differentiable tensor field which is the derivative of with respect to and thus

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection The coefficients = are differentiable functions of the considered coordinates and are called connection coefficients. Using these coefficients and the above expansion, in components, the covariant derivative of Y with respect to X can be written down as:.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection  X is called covariant derivative of X (with respect to the affine connection  ). In components we have (  Y X) i = Y j X i,j.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection define a tensor field is represented by This tensor field is symmetric in the covariant indices and is called torsion tensor field of the connection.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection The assignment of an affine connection on a differentiable manifold M is completely equivalent to the assignment of coefficients in each local coordinate system, which differentially depend on the point p and transform as under change of local coordinates.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection If M is endowed with a metric, then the manifold is called a Riemann manifold. The connection on a Riemann manifold is called Levi-Civita's affine connection. Let M be a Riemann manifold with metric locally represented by. There is exactly one affine connection  such that : (1). It is metric, i.e.,  = 0 (2). It is torsion free, i.e., T(  ) = 0.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection That is the Levi-Civita connection which is defined by the connection coefficients, called Christoffel's coefficients Consider a (pseudo) Euclidean space E n. Fixing an orthogonal Cartesian coordinate system, we can define an affine connection locally given by:.

Mathematical & Mechanical Method in Mechanical Engineering Covariant Derivative and Levi-Civita's Connection Covariant Derivative and Levi-Civita's Connection

Mathematical & Mechanical Method in Mechanical Engineering Meaning of the Covariant Derivative Meaning of the Covariant Derivative Unfortunately, there are two problems involved in the formula above: (1) What does it mean p+hY ? In general, we have not an affine structure on M and we cannot move points thorough M under the action of vectors as in affine spaces. (The reader should pay attention on the fact that affine connections and affine structures are different objects!).

Mathematical & Mechanical Method in Mechanical Engineering Meaning of the Covariant Derivative Meaning of the Covariant Derivative X(p)  T p M but X(p + hY )  T p+hY M. If something like p + hY makes sense, we expect that p + hY ≠ p because derivatives in p should investigate the behavior of the function qX(q) in a “ infinitesimal ” neighborhood of p. So the difference X(p + hY ) - X(p) does not make sense because the vectors belong to different vector spaces!.

Mathematical & Mechanical Method in Mechanical Engineering Meaning of the Covariant Derivative Meaning of the Covariant Derivative Let M be a differentiable manifold equipped with an affine connection . If X and Y are differentiable contravariant vector fields in M and p  M

Mathematical & Mechanical Method in Mechanical Engineering where  : [0,  ] → M is the unique geodesic segment referred to r starting from p with initial tangent vector Y(p) and P  [  (u),  (v)]:T  (u)  T  (v) is the vector-space isomorphism induced by the r parallel transport along a (sufficiently short) differentiable curve  : [a, b] → M for u < v and u, v  [a, b].