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Valentine Ignatov
Valentine Ignatov

Tensor Analysis on Manifolds (Dover Books on Mathematics): A Mathematical Tool for Physics and Geometry


Tensor Analysis on Manifolds (Dover Books on Mathematics) Book PDF




If you are looking for a comprehensive and accessible introduction to tensor analysis on manifolds, you might want to check out the book Tensor Analysis on Manifolds by Richard L. Bishop and Samuel I. Goldberg. This book is a classic in the field of differential geometry and provides a solid foundation for further studies in mathematics and physics. In this article, we will give you an overview of the main topics covered in the book, as well as some applications and examples of tensor analysis on manifolds. We will also provide you with a link to download the book in PDF format for free.




Tensor Analysis on Manifolds (Dover Books on Mathematics) book pdf



Introduction




Before we dive into the details of tensor analysis on manifolds, let us first review some basic concepts and definitions that will help us understand the subject better.


What is tensor analysis?




A tensor is a mathematical object that can be used to describe various physical phenomena, such as forces, stresses, strains, electric fields, magnetic fields, etc. A tensor can be thought of as a generalization of scalars, vectors, and matrices, which are special cases of tensors. A scalar is a tensor of rank zero, a vector is a tensor of rank one, and a matrix is a tensor of rank two. A tensor of rank n can be represented by an array of n indices, each ranging from 1 to some dimension d.


For example, a scalar can be written as a = a1, where the index 1 indicates that it has only one component. A vector can be written as b = bi, where i ranges from 1 to d. A matrix can be written as c = cij, where i and j range from 1 to d. A general tensor of rank n can be written as t = ti1i2...in, where each index ranges from 1 to d.


Tensors can be manipulated by various operations, such as addition, subtraction, multiplication, contraction, etc. For example, two tensors of the same rank can be added or subtracted component-wise: t + s = (ti1i2...in + si1i2...in). A tensor can be multiplied by a scalar: k * t = (k * ti1i2...in). A tensor can be contracted by summing over repeated indices: ti = t11 + t22 + ... + tdd. A tensor can also be multiplied by another tensor, resulting in a new tensor of higher or lower rank, depending on the number of indices involved.


What is a manifold?




A manifold is a mathematical space that locally resembles a Euclidean space of some dimension, but globally may have a more complicated shape or structure. For example, the surface of a sphere is a two-dimensional manifold, because every small patch of it looks like a flat plane, but the whole surface is curved and finite. A torus (donut-shaped surface) is another example of a two-dimensional manifold, which has a different topology than a sphere. A manifold can also have more than two dimensions, such as a hypersphere or a hyperboloid.


A manifold can be equipped with various structures that allow us to measure distances, angles, areas, volumes, etc. on it. For example, a metric tensor is a tensor field that assigns a positive-definite symmetric matrix to each point of the manifold, which defines the inner product and the norm of vectors at that point. A metric tensor can be used to compute the length of curves, the angle between vectors, the area of surfaces, the volume of regions, etc. on the manifold. A metric tensor also induces a notion of curvature on the manifold, which measures how much the manifold deviates from being flat.


Why study tensor analysis on manifolds?




Tensor analysis on manifolds is a powerful tool that can be used to study various aspects of geometry, topology, physics, and engineering. By using tensors, we can express quantities and equations that are invariant under coordinate transformations, which means that they do not depend on the choice of coordinates used to describe the manifold. This makes tensors suitable for describing natural phenomena that obey universal laws, such as gravity, electromagnetism, fluid dynamics, etc.


Tensor analysis on manifolds also allows us to generalize concepts and results from Euclidean spaces to more general spaces that may have curvature or other features. For example, we can define derivatives of functions and vector fields on manifolds using covariant derivatives, which take into account the connection and curvature of the manifold. We can also define integrals of functions and differential forms on manifolds using integration on chains, which generalize the notions of line integrals, surface integrals, and volume integrals.


Main Concepts and Theorems




In this section, we will briefly introduce some of the main concepts and theorems that are covered in the book Tensor Analysis on Manifolds. We will not go into too much detail or provide proofs here, but we will give some examples and intuition for each topic. For more details and rigor, we refer you to the book itself.


Tensors and tensor fields




Definition and examples




A tensor on a manifold M is an n-linear map from n copies of the tangent space TpM at some point p to the real numbers R. That is, a tensor T takes n vectors v, v, ..., v at p and produces a real number T(v, v, ..., v). The number n is called the rank or order of the tensor. A tensor field is a function that assigns a tensor to each point of the manifold.


For example, a scalar function f: M -> R is a tensor field of rank zero, because it takes zero vectors at each point and produces a real number f(p). A vector field X: M -> TM is a tensor field of rank one, because it takes one vector at each point and produces a real number X(p)(v) = , where is the inner product induced by the metric tensor. A bilinear form B: M -> T*M x T*M -> R is a tensor field of rank two, because it takes two vectors at each point and produces a real number B(p)(v,w) = B(p)(v)(w), where B(p)(v) is a linear map from TpM to R.


Operations and properties




Tensors and tensor fields can be manipulated by various operations and properties. Some of them are:


  • Addition and subtraction: Two tensor fields of the same rank can be added or subtracted component-wise. For example, if T and S are tensor fields of rank two, then (T + S)(p)(v,w) = T(p)(v,w) + S(p)(v,w) for any vectors v and w at p.



  • Scalar multiplication: A tensor field can be multiplied by a scalar function. For example, if T is a tensor field of rank two and f is a scalar function, then (f * T)(p)(v,w) = f(p) * T(p)(v,w) for any vectors v and w at p.



  • Tensor product: Two tensor fields of rank m and n can be multiplied to form a tensor field of rank m + n. For example, if T is a tensor field of rank two and S is a tensor field of rank one, then (T x S)(p)(v,w,u) = T(p)(v,w) * S(p)(u) for any vectors v, w, and u at p.



  • Contraction: A tensor field of rank n can be reduced to a tensor field of rank n - 2 by summing over repeated indices. For example, if T is a tensor field of rank four, then C(T)(p)(v,w) = Ti(p)(v,w) = T11(p)(v,w) + T22(p)(v,w) + ... + Tdd(p)(v,w) for any vectors v and w at p.



  • Covariant derivative: A tensor field of rank n can be differentiated along a vector field X to form a tensor field of rank n + 1. For example, if T is a tensor field of rank two and X is a vector field, then DXT(p)(v,w) = X(p)(T(p)(v,w)) - T(p)(DXv,w) - T(p)(v,DXw) for any vectors v and w at p. The covariant derivative takes into account the connection and curvature of the manifold.



  • Lie derivative: A tensor field of rank n can be differentiated along the flow of a vector field X to form another tensor field of rank n. For example, if T is a tensor field of rank two and X is a vector field, then LXT(p)(v,w) = limt->0 (1/t) * (T(φ(p))(Dφ(p)v,Dφ(p)w) - T(p)(v,w)) for any vectors v and w at p. The Lie derivative measures the change of the tensor along the direction of X.



Differential forms and exterior algebra




Definition and examples




A differential form on a manifold M is an alternating multilinear map from the tangent space TpM at some point p to the real numbers R. That is, a differential form ω takes k vectors v, v, ..., v at p and produces a real number ω(v, v, ..., v). The number k is called the degree or order of the differential form. A differential form is alternating if it changes sign when two arguments are swapped: ω(v, ..., v, ..., v, ...) = -ω(v, ..., v, ..., v, ...). A differential form field is a function that assigns a differential form to each point of the manifold.


For example, a scalar function f: M -> R is a differential form field of degree zero, because it takes zero vectors at each point and produces a real number f(p). A one-form or covector field α: M -> T*M is a differential form field of degree one, because it takes one vector at each point and produces a real number α(p)(v) = , where is the dual pairing between T*M and TM. A two-form or bivector field β: M -> ΛT*M is a differential form field of degree two, because it takes two vectors at each point and produces a real number β(p)(v,w) = , where x is the wedge product between one-forms and vectors.


Operations and properties




Differential forms and differential form fields can be manipulated by various operations and properties. Some of them are:


  • Addition and subtraction: Two differential form fields of the same degree can be added or subtracted component-wise. For example, if ω and η are differential form fields of degree two, then (ω + η)(p)(v,w) = ω(p)(v,w) + η(p)(v,w) for any vectors v and w at p.



  • Scalar multiplication: A differential form field can be multiplied by a scalar function. For example, if ω is a differential form field of degree two and f is a scalar function, then (f * ω)(p)(v,w) = f(p) * ω(p)(v,w) for any vectors v and w at p.



  • Wedge product: Two differential form fields of degree k and l can be multiplied to form a differential form field of degree k + l. For example, if ω is a differential form field of degree two and α is a differential form field of degree one, then (ω ^ α)(p)(v,w,u) = ω(p)(v,w) * α(p)(u) - ω(p)(v,u) * α(p)(w) + ω(p)(w,u) * α(p)(v) for any vectors v, w, and u at p. The wedge product is antisymmetric and associative.



  • Exterior derivative: A differential form field of degree k can be differentiated to form a differential form field of degree k + 1. For example, if ω is a differential form field of degree two and X and Y are vector fields, then dω(p)(X,Y,Z) = X(p)(ω(p)(Y,Z)) - Y(p)(ω(p)(X,Z)) + Z(p)(ω(p)(X,Y)) - ω(p)([X,Y],Z) + ω(p)([X,Z],Y) - ω(p)([Y,Z],X) for any vectors X, Y, and Z at p. The exterior derivative is linear and satisfies the Leibniz rule and the Poincaré lemma.



  • Interior product: A differential form field of degree k can be contracted with a vector field X to form a differential form field of degree k - 1. For example, if ω is a differential form field of degree two and X is a vector field, then iXω(p)(v) = ω(p)(X,v) for any vector v at p. The interior product is linear and satisfies the Leibniz rule and the Cartan formula.



  • Integration: A differential form field of degree n can be integrated over an n-dimensional oriented manifold M to produce a real number. For example, if ω is a differential form field of degree two and S is a two-dimensional oriented surface in M, then Sω = Sω(v,w)dA(v,w), where v and w are tangent vectors to S at each point and dA(v,w) is the area element induced by the metric tensor. The integration is linear and satisfies the Stokes' theorem.



Integration on manifolds




Definition and examples




Integration on manifolds is a generalization of the classical notions of line integrals, surface integrals, and volume integrals to arbitrary dimensions and shapes. Integration on manifolds allows us to compute the total amount of some quantity over a region or along a curve on a manifold. For example, we can compute the mass, the charge, the flux, the circulation, the work, etc. of various physical systems using integration on manifolds.


To define integration on manifolds, we need some additional concepts and tools. First, we need to define what is a manifold with boundary, which is a manifold that has some boundary points where it ends or where it has sharp edges or corners. For example, a disk or a hemisphere are manifolds with boundary, while a sphere or a torus are manifolds without boundary. Second, we need to define what is an orientation on a manifold, which is a consistent choice of direction or sign for each point of the manifold. For example, an orientation on a surface can be given by choosing a normal vector or an outward direction at each point. Third, we need to define what is a chain on a manifold, which is a formal sum of oriented submanifolds with coefficients in Z (the integers). For example, a chain on a surface can be given by adding or subtracting oriented curves or regions on the surface. Fourth, we need to define what is a boundary operator on a chain, which is a function that maps a chain of dimension k to a chain of dimension k - 1 by taking the boundary of each submanifold in the chain. For example, the boundary of a curve is the sum of its endpoints, and the boundary of a region is the sum of its edges.


With these concepts and tools, we can define integration on manifolds as follows: Let M be an n-dimensional oriented manifold with boundary, and let ω be an n-form on M. Then, the integral of ω over M is given by Mω = iciMiω, where Mi is a finite collection of oriented submanifolds of M that cover M, and ci are integers such that (iciMi) = M. Here, M is the boundary of M as a chain, and (iciMi) is the boundary of the chain iciMi. The integral of ω over M does not depend on the choice of Mi and ci, as long as they satisfy the above conditions.


Operations and properties




Integration on manifolds can be manipulated by various operations and properties. Some of them are:


  • Linearity: The integral of a linear combination of n-forms is equal to the linear combination of the integrals of the n-forms. For example, if ω and η are n-forms on M and f and g are scalar functions, then M(f * ω + g * η) = f * Mω + g * Mη.



  • Additivity: The integral of an n-form over a union of disjoint submanifolds is equal to the sum of the integrals of the n-form over each submanifold. For example, if M = M1 M2 and M1 M2 = , then Mω = M1ω + M2ω.



  • Invariance: The integral of an n-form over an oriented manifold is invariant under orientation-preserving diffeomorphisms. For example, if φ: M -> N is an orientation-preserving diffeomorphism and ω is an n-form on N, then M(φ*ω) = Nω, where φ*ω is the pullback of ω by φ.



  • Change of variables: The integral of an n-form over an oriented manifold can be computed by changing coordinates using a smooth map. For example, if ψ: U -> V is a smooth map between open subsets of R, and ω is an n-form on V, then U(ψ*ω) = Vω, where ψ*ω is the pullback of ω by ψ.



  • Stokes' theorem: The integral of an exterior derivative of a (n-1)-form over an oriented manifold with boundary is equal to the integral of the (n-1)-form over the boundary of the manifold. For example, if M is an oriented manifold with boundary M and ω is a (n-1)-form on M, then Mdω = Mω, where dω is the exterior derivative of ω.



Riemannian geometry and curvature




Definition and examples




Riemannian geometry is a branch of differential geometry that studies manifolds that are equipped with a metric tensor, which defines the inner product and the norm of vectors at each point. A metric tensor also induces a notion of distance, angle, area, volume, etc. on the manifold. A Riemannian manifold is a manifold with a metric tensor.


A metric tensor also induces a notion of curvature on the manifold, which measures how much the manifold deviates from being flat. Curvature can be defined in various ways, such as sectional curvature, Ricci curvature, scalar curvature, etc. Curvature can also be expressed in terms of tensors, such as the Riemann curvature tensor, the Ricci curvature tensor, the scalar curvature function, etc.


For example, a flat plane is a Riemannian manifold with zero curvature everywhere. A sphere is a Riemannian manifold with positive constant curvature everywhere. A hyperbolic plane is a Riemannian manifold with negative constant curvature everywhere. A cylinder is a Riemannian manifold with zero curvature along one direction and positive constant curvature along another direction.


Operations and properties




Riemannian geometry and curvature can be manipulated by various operations and properties. Some of them are:


  • Geodesics: A geodesic on a Riemannian manifold is a curve that locally minimizes the distance between its endpoints. For example, a straight line on a flat plane, a great circle on a sphere, and a hypercycle on a hyperbolic plane are geodesics. Geodesics can be characterized by the property that they have zero acceleration or zero covariant derivative along themselves.



  • Parallel transport: Parallel transport on a Riemannian manifold is a way of moving a vector along a curve on the manifold without changing its direction or length. For example, parallel transporting a vector along a straight line on a flat plane does not change the vector at all, but parallel transporting a vector along a great circle on a sphere may change the vector depending on the angle of rotation. Parallel transport can be defined by the property that it preserves the inner product between vectors or that it has zero covariant derivative along the curve.



  • Levi-Civita connection: The Levi-Civita connection on a Riemannian manifold is a connection that is compatible with the metric tensor and torsion-free. That is, it satisfies the following properties: DX = XY,Z> + XZ> and DXY - DYX - [X,Y] = 0 for any vector fields X, Y, and Z. The Levi-Civita connection can be used to define the covariant derivative, the parallel transport, and the geodesics on the manifold.



  • Riemann curvature tensor: The Riemann curvature tensor on a Riemannian manifold is a tensor field of rank four that measures the change of parallel transport around an infinitesimal loop on the manifold. For example, if X and Y are vector fields and Z is a vector at some point p, then R(X,Y)Z = DXDYZ - DYDXZ - D[X,Y]Z is the difference between parallel transporting Z along X and then Y versus parallel transporting Z along Y and then X. The Riemann curvature tensor can also be expressed in terms of the commutator of covariant derivatives: R(X,Y)Z = [DX,DY]Z.



Ricci curvature tensor: The Ricci curvature tensor on a Riemannian manifold is a tensor fiel


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