Riemannian geometry

Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




  • (shape modality \dashv flat modality \dashv sharp modality)

    (ʃ)(ʃ \dashv \flat \dashv \sharp )

  • dR-shape modality\dashv dR-flat modality

    ʃ dR dRʃ_{dR} \dashv \flat_{dR}

  • tangent cohesion

    • differential cohomology diagram
    • differential cohesion

      • (reduction modality \dashv infinitesimal shape modality \dashv infinitesimal flat modality)

        (&)(\Re \dashv \Im \dashv \&)

      • graded differential cohesion

        • fermionic modality\dashv bosonic modality \dashv rheonomy modality

          (Rh)(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)

        • id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous *

          \array{ && id &\dashv& id \ && \vee && \vee \ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \ && \bot && \bot \ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \ && \vee && \vee \ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \ && \bot && \bot \ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \ && \vee && \vee \ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \ && \bot && \bot \ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \ && \vee && \vee \ && \emptyset &\dashv& \ast }



          Lie theory, ∞-Lie theory

          differential equations, variational calculus

          Chern-Weil theory, ∞-Chern-Weil theory

          Cartan geometry (super, higher)



          On a Riemannian manifold (X,g)(X,g), a geodesic (or geodesic line, geodesic path) is a path x:IXx : I \to X, for some (possibly infinite) interval II, which locally minimizes distance.


          One way to define a geodesic is as a critical point of the functional of length

          (x:IM) Ig(x(t),x(t))dt (x : I \to M) \mapsto \int_I \sqrt{g(\stackrel{\cdot}{x}(t),\stackrel{\cdot}{x}(t))} dt

          on the appropriate space of curves IXI \to X, where gg is the metric tensor. This implies that it satisfies the corresponding Euler-Lagrange equations, which in this case means that the covariant derivative (for the Levi-Civita connection)

          xx=0 \nabla_{\stackrel{\cdot}{x}} \stackrel{\cdot}{x} = 0


          In local coordinates, with Christoffel symbols Γ jk i\Gamma^i_{jk} the Euler-Lagrange equations for geodesics form a system

          d 2x idt 2+ jkΓ jk i(x)dx jdtdx kdt=0. \frac{d^2 x^i}{dt^2} + \sum_{jk} \Gamma^i_{jk}(x) \frac{d x^j}{dt} \frac{d x^k}{dt} = 0.

          So this means that a curve is a geodesic if at every point its tangent vector is the parallel transport of the tangent vector at the start point along the curve.

          Minimizing geodesics

          A geodesic may not globally minimize the distance between its end points. For instance, on a 2-dimensional sphere, geodesics are arcs of great circle?s. Any two distinct non-antipodal points are connected by exactly two such geodesics, one shorter than the other (you can go from Los Angeles to Boston directly across North America, or the long way around the world).

          A geodesic which does globally minimize distance between its end points is called a minimizing geodesic. The length of a minimizing geodesic between two points defines a distance function for any Riemannian manifold which makes it into a metric space.

          Last revised on December 10, 2014 at 16:39:22. See the history of this page for a list of all contributions to it.