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Riemannian Geometry

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Riemannian Geometry Riemannian Geometry it is a draft of Lecture Notes of H.M. Khudaverdian. Manchester, 20-th May 2011 Contents 1 Riemannian manifolds 4 1.1 Manifolds. Tensors. (Recalling) . 4 1.2 Riemannian manifold|manifold equipped with Riemannian metric . 7 1.2.1 ∗ Pseudoriemannian manifold . 11 1.3 Scalar product. Length of tangent vectors and angle between vectors. Length of the curve . 11 1.3.1 Length of the curve . 12 1.4 Riemannian structure on the surfaces embedded in Euclidean space . 15 1.4.1 Internal and external coordinates of tangent vector . 15 1.4.2 Explicit formulae for induced Riemannian metric (First Quadratic form) . 17 1.4.3 Induced Riemannian metrics. Examples. 20 1.4.4 ∗Induced metric on two-sheeted hyperboloid embedded in pseudo-Euclidean space. 27 1.5 Isometries of Riemanian manifolds. 28 1.5.1 Examples of local isometries . 29 1.6 Volume element in Riemannian manifold . 31 1.6.1 Volume of parallelepiped . 31 1.6.2 Invariance of volume element under changing of coor- dinates . 33 1.6.3 Examples of calculating volume element . 34 1 2 Covariant differentiaion. Connection. Levi Civita Connec- tion on Riemannian manifold 36 2.1 Differentiation of vector field along the vector field.—Affine connection . 36 2.1.1 Definition of connection. Christoffel symbols of con- nection . 37 2.1.2 Transformation of Christoffel symbols for an arbitrary connection . 39 2.1.3 Canonical flat affine connection . 40 2.1.4 ∗ Global aspects of existence of connection . 43 2.2 Connection induced on the surfaces . 44 2.2.1 Calculation of induced connection on surfaces in E3. 45 2.3 Levi-Civita connection . 47 2.3.1 Symmetric connection . 47 2.3.2 Levi-Civita connection. Theorem and Explicit formulae 47 2.3.3 Levi-Civita connection on 2-dimensional Riemannian manifold with metric G = adu2 + bdv2. 49 2.3.4 Example of the sphere again . 49 2.4 Levi-Civita connection = induced connection on surfaces in E3 50 3 Parallel transport and geodesics 52 3.1 Parallel transport . 52 3.1.1 Definition . 52 3.1.2 ∗Parallel transport is a linear map . 53 3.1.3 Parallel transport with respect to Levi-Civita connection 54 3.2 Geodesics . 54 3.2.1 Definition. Geodesic on Riemannian manifold. 54 3.2.2 Un-parameterised geodesic . 55 3.2.3 Geodesics on surfaces in E3 . 56 3.3 Geodesics and Lagrangians of "free" particle on Riemannian manifold. 58 3.3.1 Lagrangian and Euler-Lagrange equations . 58 3.3.2 Lagrangian of "free" particle . 58 3.3.3 Equations of geodesics and Euler-Lagrange equations . 59 3.3.4 Examples of calculations of Christoffel symbols and geodesics using Lagrangians. 60 3.3.5 Variational principe and Euler-Lagrange equations . 62 3.4 Geodesics and shortest distance. 63 2 3.4.1 Again geodesics for sphere and Lobachevsky plane . 65 4 Surfaces in E3 67 4.1 Formulation of the main result. Theorem of parallel transport over closed curve and Theorema Egregium . 67 4.1.1 GaußTheorema Egregium . 69 4.2 Derivation formulae . 70 4.2.1 ∗Gauss condition (structure equations) . 72 4.3 Geometrical meaning of derivation formulae. Weingarten op- erator and second quadratic form in terms of derivation for- mulae. 73 4.3.1 Gaussian and mean curvature in terms of derivation formulae . 75 4.4 Examples of calculations of derivation formulae and curvatures for cylinder, cone and sphere . 76 4.5 ∗Proof of the Theorem of parallel transport along closed curve. 80 5 Curvtature tensor 84 5.1 Definition . 84 5.1.1 Properties of curvature tensor . 85 5.2 Riemann curvature tensor of Riemannian manifolds. 86 5.3 yCurvature of surfaces in E3.. Theorema Egregium again . 87 5.4 Relation between Gaussian curvature and Riemann curvature tensor. Straightforward proof of Theorema Egregium . 88 5.4.1 ∗Proof of the Proposition (5.25) . 90 5.5 Gauss Bonnet Theorem . 94 6 Appendices 97 6.1 ∗Integrals of motions and geodesics. 97 6.1.1 ∗Integral of motion for arbitrary Lagrangian L(x; x_) . 97 6.1.2 ∗Basic examples of Integrals of motion: Generalised momentum and Energy . 97 6.1.3 ∗Integrals of motion for geodesics . 98 6.1.4 ∗Using integral of motions to calculate geodesics . 100 6.2 Induced metric on surfaces. 101 6.2.1 Recalling Weingarten operator . 101 6.2.2 Second quadratic form . 102 6.2.3 Gaussian and mean curvatures . 103 3 6.2.4 Examples of calculation of Weingarten operator, Sec- ond quadratic forms, curvatures for cylinder, cone and sphere. 103 1 Riemannian manifolds 1.1 Manifolds. Tensors. (Recalling) I recall briefly basics of manifolds and tensor fields on manifolds. An n-dimensional manifold is a space such that in a vicinity of any point one can consider local coordinates fx1; : : : ; xng (charts). One can consider different local coordinates. If both coordinates fx1; : : : ; xng, fy1; : : : ; yng are defined in a vicinity of the given point then they are related by bijective transition functions (functions defined on domains in Rn and taking values in Rn). 8 > 10 10 1 n >x = x (x ; : : : ; x ) > 0 0 <>x2 = x2 (x1; : : : ; xn) >::: > − 0 − 0 >xn 1 = xn 1 (x1; : : : ; xn) :> 0 0 xn = xn (x1; : : : ; xn) We say that manifold is differentiable or smooth if transition functions are diffeomorphisms, i.e. they are smooth and rank of Jacobian is equal to k, i.e. 0 0 0 0 1 @x1 @x1 @x1 1 2 ::: n B @x 0 @x 0 @x 0 C B @x2 @x2 ::: @x2 C det B @x1 @x2 @xn C =6 0 (1.1) @ ::: A 0 0 0 @xn @xn @xn @x1 @x2 ::: @xn A good example of manifold is an open domain D in n-dimensional vector space Rn. Cartesian coordinates on Rn define global coordinates on D. On the other hand one can consider an arbitrary local coordinates in different domains in Rn. E.g. one can consider polar coordinates fr; 'g in a domain D = fx; y : y > 0g of R2 (or in other domain of R2) defined by standard formulae: ( x = r cos ' ; y = r sin ' 4 ! ( ) @x @x − @r @' cos ' r sin ' det @y @y = det = r (1.2) @r @' sin ' r cos ' or one can consider spherical coordinates fr; θ; 'g in a domain D = fx; y; z : x > 0; y > 0; z > 0g of R3 (or in other domain of R3) defined by standard for- mulae 8 <>x = r sin θ cos ' y = r sin θ sin ' ; :> z = r cos θ , 0 1 0 1 @x @x @x − B @r @θ @' C sin θ cos ' r cos θ cos ' r sin θ sin ' @ @y @y @y A @ A 2 det @r @θ @' = det sin θ sin ' r cos θ sin ' r sin θ cos ' = r sin θ @z @z @z cos θ −r sin θ 0 @r @θ @' (1.3) Choosing domain where polar (spherical) coordinates are well-defined we have to be award that coordinates have to be well-defined and transition functions (1.1) have to be diffeomorphisms. Examples of manifolds: Rn, Circle S1, Sphere S2, in general sphere Sn, torus S1 × S1, cylinder, cone, . We also have to recall briefly what are tensors on manifold. Tensors on Manifold For every point p on manifold M one can consider tangent vector space TpM|the space of vectors tangent to the manifold at the point M. i @ Tangent vector A = A @xi . Under changing of coordinates it transforms as follows: m0 @ @x @ 0 @ A = Ai = Ai = Am @xi @xi @xm0 @xm0 Hence i0 0 @x Ai = Ai (1.4) @xi ∗ Consider also cotangent space TpM (for every point p on manifold M)| space of linear functions on tangent vectors, i.e. space of 1-forms which sometimes are called covectors.: 5 i One-form (covector) ! = !idx transforms as follows m m @x m0 m0 ! = ! dx = ! dx = ! 0 dx : m m @xm0 m Hence @xm ! 0 = ! : (1.5) m @xm0 m Tensors: One can consider contravariant tensors of the rank p @ @ @ T = T i1i2:::ip ⊗ ⊗ · · · ⊗ @xi1 @xi2 @xik with components fT i1i2:::ik g. One can consider covariant tensors of the rank q j1 ⊗ j2 ⊗ jq S = Sj1j2:::jq dx dx : : : dx f g with components Sj1j2:::jq . One can also consider mixed tensors: i i :::i @ @ @ 1 2 p ⊗ ⊗ · · · ⊗ ⊗ j1 ⊗ j2 ⊗ jq Q = Qj j :::j dx dx : : : dx 1 2 q @xi1 @xi2 @xik ( ) p f i1i2:::ip g with components Qj j :::j . We call these tensors tensors of the type . (1 )2 q q p Tensors of the type are called contravariant tensors of the rank p. 0 ( ) 0 Tensors of the type are called covariant tensors of the rank q. q Having in mind( (1.4)) and (1.5) we come to the rule of transformation for p tensors of the type : q 0 0 0 0 0 0 i1 i2 ip j1 j2 jq i1i2:::ip @x @x @x @x @x @x i1i2:::ip Q 0 0 0 = ::: 0 0 ::: 0 Q (1.6) j j :::j i i i j j j j1j2:::jq 1 2 q @x 1 @x 2 @x p @x 1 @x 2 @x q ( ) p E.g. if S is a covariant tensor of rank 2 (tensor of the type ) then ik q @xi @xk S 0 0 = S : (1.7) i k @xi0 @xk0 ik 6 ( ) 1 If Ai is a tensor of rank (linear operator on T M) then k 1 p i0 k i0 @x @x i A 0 = A k @xi @xk0 k Remark Transformations formulae (1.4)|(1.7) define vectors, covectors and in generally any tensor fields in components.
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