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Riemannian Geometry

1 Basic Things

  • Convention 1.1. \(R_{XY} = [\nabla _Y,\nabla _X]\) if \([X,Y] = 0\).

  • Definition 1.2. The energy \(R(\gamma )\) of a smooth curve \(\gamma \) is \(\int _{\gamma } \Vert \dot {\gamma }(t)\Vert ^2\).

  • Lemma 1.3 (First Variation Formula). Let \(V\) be a variational field of a curve \(\gamma \) defined on \([0,1]\) that is piecewise smooth on the subdivision \(0 = t_0<t_1<\dots <t_n=1\).

    \[\frac 1 2dE(V) = \sum _i \langle V(t_i),\Delta \gamma '(t_i)\rangle - \int \langle V,\gamma ''\rangle \]

  • Proof. Let \(\gamma _s\) be a variation. Then we can calculate:

    \[\frac 1 2\frac d {ds} \int \langle \gamma _s',\gamma _s'\rangle = \int \langle \frac D {ds} \frac d {dt} \gamma _s,\gamma _s'\rangle = \int \langle \frac D {dt} V,\gamma '\rangle \]

    \[=\int \frac d {dt} \langle V ,\gamma '\rangle - \int \langle V,\gamma ''\rangle \]

    And finally, use the fundamental theorem of calculus on the first term. □

  • Lemma 1.4 (Second Variation Formula). Let \(V,W\) be endpoint-fixing variations of a geodesic \(\gamma \) that are piecewise smooth (and continuous). Then if \(H\) is the Hessian of the energy functional, \(\frac 1 2H(V,W) = \sum _i \langle V,\Delta \frac D{dt} W(t_i) \rangle + \int \langle R_{\gamma ',W}\gamma ' - W'',V\rangle \)

  • Proof. Let \(\gamma \) be a two parameter variation in the variables \(s_1,s_2\). Then we can calculate:

    \[\frac d {d{s_2}} \int \langle \frac D {dt} \frac d {ds_1} \gamma , \gamma '\rangle = \int \langle \frac D {ds_2} \frac D {dt}\frac d {ds_1} \gamma , \gamma '\rangle + \int \langle \frac D {dt}\frac d {ds_1} \gamma , \frac D {ds_2} \gamma '\rangle \]

    For the first term, we get

    \[ = \int \langle \frac D {dt} \frac D {ds_2}\frac d {ds_1} \gamma , \gamma '\rangle + \int \langle R_{W,\gamma '}V, \gamma '\rangle = \int \frac d {dt} \langle \frac D {ds_2}\frac d {ds_1} \gamma , \gamma '\rangle + \int \langle R_{\gamma ,W}\gamma ',V\rangle \]

    Now we can calculate

    \[ \int \frac d {dt} \langle \frac D {ds_2}\frac d {ds_1} \gamma , \gamma '\rangle = \int \frac d {dt}\frac d {ds_2} \langle \frac d {ds_1} \gamma , \gamma '\rangle - \int \frac d {dt}\langle V,\frac D {dt} W\rangle \]

    The first part is \(0\) because the variations fix the ends, and the second is the sum in the answer. Finally we calculate

    \[\int \langle \frac D {dt}\frac d {ds_1} \gamma , \frac D {ds_2} \gamma '\rangle = \int \frac d {dt} \langle V \gamma , \frac D {ds_2} \gamma '\rangle - \int \langle V, W''\rangle \]

    Where the first term is \(0\) again. □

  • Lemma 1.5. \(\Hess (u)(X,Y) =\langle \nabla _Y\nabla u,X\rangle = YXu-\nabla _X(Y)u\)

  • Lemma 1.6 (Bochner’s Formula). \(\frac 1 2 \Delta |\nabla u |^2 = |\Hess u|^2 + \nabla u (\Delta u) + \Ric (\nabla u, \nabla u)\).

  • Proof. In normal coordinates of a point \(p\) the \(\partial _i\) commute with each other and have \(\nabla _{\partial _i}\partial _j(p) = 0\). Then we can compute:

    \[\frac 1 2 \Delta |\nabla u|^2 = \partial _i\langle \nabla _{\partial _i}\nabla u, \nabla u\rangle = \partial _i \Hess (u)(\partial _i,\nabla u) = \partial _i \langle \nabla _{\nabla u}\nabla u, \partial _i\rangle \]

    \[ = \langle \nabla _{\partial _i}\nabla _{\nabla u} \nabla u, \partial _i \rangle = \langle \nabla _{\nabla u} \nabla _{\partial _i}\nabla u, \partial _i \rangle + \langle R_{\nabla u, \partial _i} \nabla u, \partial _i \rangle + \langle \nabla _{[\partial _i,\nabla u]}\nabla u, \partial _i \rangle \]

    The second term is \(\Ric (\nabla u, \nabla u)\).

    We can compute

    \[|\Hess u|^2 = \Hess u(\partial _i,\partial _j) \langle \nabla _{\partial _i} \nabla u, \partial _j\rangle = \Hess u(\partial _i, \nabla _{\partial _i}\nabla u)\]

    \[= \langle \nabla _{[\partial _i,\nabla u]}\nabla u, \partial _i\rangle - \langle \nabla _{\partial _i} \nabla u, \nabla _{\nabla u}\partial _i\rangle \]

    And also

    \[\nabla u (\Delta u) = \nabla u \langle \nabla _{\partial _i} \nabla u, \partial _i\rangle =\langle \nabla _{\nabla u} \nabla _{\partial _i} \nabla u, \partial _i\rangle + \langle \nabla _{\partial _i} \nabla u,\nabla _{\nabla u} \partial _i\rangle \]

    So these add up to the first and third term. □