Abstract

Given a Riemannian manifold $\calM$, the Sasaki metric is a natural Riemannian metric for the tangent bundle $\T\calM$. It is often described axiomatically, but it also admits a simple, transparent formula for the common case where $\calM$ is a Riemannian submanifold of a Euclidean space. This post provides the formula and illustrates it on the sphere.

If $\calM$ is a smooth manifold, then its tangent bundle \[\T\calM = \{ (x, v) : x \in \calM \textrm{ and } v \in \T_x\calM \}\] is a smooth manifold too, of twice the dimension. Give $\calM$ a Riemannian metric, and it is natural to ask: can we give $\T\calM$ a Riemannian metric that somehow interacts nicely with the one on $\calM$? Several proposals have been made, with the Sasaki metric emerging as one of the best known (Sasaki 1958).

A look at the Wikipedia page reveals an axiomatic definition (we’ll get to it later): the Sasaki metric on $\T\calM$ is the only one that meets three (arguably natural) desiderata for how it relates to the metric on $\calM$.

For computations and intuition, though, it is often convenient to have an actual formula for the metric. This post works out an easily computable expression for the case where $\calM$ is a Riemannian submanifold of a Euclidean space, then provides an example and some context for how one might have landed on that metric in the first place.

TL;DR

Let $\calE$ be a Euclidean space, with inner product $\inner{\cdot}{\cdot}_\calE$. Let $\calM$ be an embedded submanifold of $\calE$ (no metric yet). In particular, $\calM$ is a subset of $\calE$, and each tangent space of $\calM$ is a linear subspace of $\calE$. The tangent bundle \[ \T\calM = \{ (x, v) \in \calE \times \calE : x \in \calM \textrm{ and } v \in \T_x \calM \} \] is an embedded submanifold of $\calE \times \calE$. Thus, $\T_{(x, v)} \T\calM$ (the tangent space to $\T\calM$ at $(x, v) \in \T\calM$) is a linear subspace of $\calE \times \calE$.

Let $\Proj_x$ denote the orthogonal projector from $\calE$ to $\T_x \calM$—this is with respect to the Euclidean metric on $\calE$.

Give $\calM$ a Riemannian metric $\inner{\cdot}{\cdot}_x$ (an arbitrary one for now). The following expression defines a Riemannian metric for $\T\calM$: \[\begin{align} \inner{(\dot x_1, \dot v_1)}{(\dot x_2, \dot v_2)}_{(x, v)} & = \inner{\dot x_1}{\dot x_2}_x + \inner{\Proj_x \dot v_1}{\Proj_x \dot v_2}_x. \end{align} \qquad(1)\] Indeed, we shall see that $\dot x_i$ is in $\T_x \calM$; moreover, the proposed expression is clearly smooth, bilinear and symmetric. One also checks that it is positive definite because $\inner{(\dot x, \dot v)}{(\dot x, \dot v)}_{(x, v)} = \|\dot x\|_x^2 + \|\Proj_x \dot v\|_x^2 \geq 0$ is zero if and only if $(\dot x, \dot v) = 0$ (see the margin note).

For $\|\dot x\|_x^2 + \|\Proj_x \dot v\|_x^2$ to be zero, you need $\dot x = 0$ and $\Proj_x \dot v = 0$, but $\dot x = 0$ implies $\Proj_x \dot v = \dot v$ owing to Eq. 3 below, so $\dot v = 0$.

If $\calM$ is a Riemannian submanifold of $\calE$, meaning $\inner{\cdot}{\cdot}_x$ is simply $\inner{\cdot}{\cdot}_\calE$ restricted to the tangent space at $x$, then Eq. 1 is the Sasaki metric for $\T\calM$, inherited from $\calM$.

In that case, we can further simplify the expression to \[\begin{align} \inner{(\dot x_1, \dot v_1)}{(\dot x_2, \dot v_2)}_{(x, v)} & = \inner{\dot x_1}{\dot x_2}_\calE + \inner{\dot v_1}{\Proj_x \dot v_2}_\calE \end{align} \qquad(2)\] since $\Proj_x$ is an orthogonal projector in $\calE$.

Example: Sasaki metric for the sphere

Let’s run through an example, then we’ll redo it step-by-step to illustrate the general procedure.

Let $\Sn = \{ x \in \Rn : x^\top x = 1 \}$ be the sphere as a Riemannian submanifold of $\Rn$ with the usual metric $\inner{u}{v}_\calE = u^\top v$. Then, the tangent bundle is \[ \T\Sn = \{ (x, v) \in \Rn \times \Rn : x^\top x = 1 \textrm{ and } x^\top v = 0 \} \] and the tangent space to that tangent bundle at $(x, v)$ (as a subspace of $\Rn \times \Rn$) is \[ \T_{(x, v)}\T\Sn = \{ (\dot x, \dot v) \in \Rn \times \Rn : x^\top \dot x = 0 \textrm{ and } \dot x^\top v + x^\top \dot v = 0 \}. \] Since the orthogonal projector to $\T_x \Sn$ is $\Proj_x(z) = z - (x^\top z) x$, the Sasaki metric on $\T\Sn$ is \[ \inner{(\dot x_1, \dot v_1)}{(\dot x_2, \dot v_2)}_{(x, v)} = \dot x_1^\top \dot x_2^{} + \dot v_1^\top \dot v_2^{} - (x^\top \dot v_1)(x^\top\dot v_2). \] In particular, this is different from the Riemannian submanifold metric one would get from looking at $\T\Sn$ as a submanifold of $\calE \times \calE$ (that would only have the first two terms).

Getting there from a general formula

While the Sasaki metric is typically described axiomatically, Musso and Tricerri (1988) also provide a direct expression for the general case. Let’s particularize that formula (stated after some context) in order to get to Eq. 1.

Since $\T\calM$ has twice the dimension of $\calM$, and since we want to use the metric on $\calM$ to build a metric on $\T\calM$, it makes sense that we should try to “extract” two tangent vectors of $\calM$ for each tangent vector of $\T\calM$, to then push through the metric of $\calM$.

Let’s see how this might work.

As a running example, think of the sphere $\Sn = \{ x \in \Rn : x^\top x = 1 \}$ embedded in $\Rn$. Then, \[\T\Sn = \{(x, v) \in \Rn \times \Rn : x^\top x = 1 \textrm{ and } x^\top v = 0\}.\]

Fix a point $(x, v)$ on $\T\calM$, so that $x$ is a point on $\calM$ and $v$ is a tangent vector at $x$.

Consider a smooth curve $t \mapsto c(t)$ on $\T\calM$ that passes through $(x, v)$ at time $t = 0$. That is, $c(t) = (x(t), v(t))$ is smooth, with the properties that \[\begin{align*} x(0) = x, && v(0) = v && \textrm{ and } && v(t) \in \T_{x(t)}\calM \textrm{ for all } t. \end{align*}\] By definition, the velocity of $c$ as it passes through $(x, v)$, that is, $c'(0) = (x'(0), v'(0))$, is a tangent vector to $\T\calM$ at $(x, v)$—and all tangent vectors are of that form.

Here, the “prime” denotes a standard derivative: $c'(t) = \ddt c(t)$. This makes sense because $x$ and $v$ live in the Euclidean embedding space too (they map $t$ to an element of $\calE$): we can just differentiate there. Thus, \[\begin{multline*} \T_{(x, v)} \T\calM = \big\{ (\dot x, \dot v) \in \calE \times \calE : \\ \textrm{there exists a smooth curve $c$ on } \T\calM \textrm{ with } c'(0) = (\dot x, \dot v) \big\}. \end{multline*}\]

For the sphere, $x(t) \in \Sn$ means $x(t)^\top x(t) = 1$ for all $t$. Likewise, $v(t) \in \T_{x(t)}\Sn$ means $x(t)^\top v(t) = 0$ for all $t$. Differentiating reveals $x(t)^\top x'(t) = 0$ and $x'(t)^\top v(t) + x(t)^\top v'(t) = 0$ for all $t$. Evaluate at $t = 0$ and use $x(0) = x$, $v(0) = v$, $x'(0) = \dot x$ and $v'(0) = \dot v$ to obtain \[ \T_{(x, v)} \T\Sn = \{ (\dot x, \dot v) \in \Rn \times \Rn : x^\top \dot x = 0 \textrm{ and } \dot x^\top v + x^\top \dot v = 0 \}. \] Notice how $\dot x$ is tangent to $\Sn$ at $x$, but not necessarily so for $\dot v$.

Can we relate $(\dot x, \dot v)$ to tangent vectors of $\calM$? Well, $t \mapsto x(t)$ is a curve on $\calM$ and $x(0) = x$, so by definition $\dot x = x'(0)$ is a tangent vector to $\calM$ at $x$. However, $\dot v = v'(0)$ may not be tangent at $x$.

That said, $t \mapsto v(t)$ is a smooth vector field along the curve $t \mapsto x(t)$. Since $\calM$ is a Riemannian manifold, there is a preferred way to differentiate $v$, namely, the covariant derivative $t \mapsto \Ddt v(t)$ induced by the Riemannian connection (as opposed to the standard derivative $t \mapsto v'(t)$ in the embedding space). Also, $\Ddt v$ is itself a vector field along the curve $x$, so that $\Ddt v(0)$ in particular is a tangent vector to $\calM$ at $x(0) = x$.

Thus, we see that $(\dot x, \dot v) = c'(0)$ (an arbitrary tangent vector to $\T\calM$ at $(x, v)$) can be mapped to two tangent vectors to $\calM$ at $x$: \[\begin{align*} u := x'(0) && \textrm{ and } && w := \Ddt v(0). \end{align*}\] At this stage, we might worry that $u, w \in \T_x \calM$ depend not only on $(\dot x, \dot v)$ but also on our choice of curve $c$. That is not the case, but let’s skip this concern, for it will dissolve entirely in our special case of interest momentarily.

For the sphere, pick some $(\dot x, \dot v) \in \T_{(x, v)}\T\Sn$, and any associated curve $c$ on $\T\Sn$ so that $c(0) = (x, v)$ and $c'(0) = (x'(0), v'(0)) = (\dot x, \dot v)$. We always have \[u = \dot x.\] As for $w$, we need to compute $\Ddt v(0)$. This depends on the Riemannian metric we choose for $\Sn$. The usual metric is that which makes $\Sn$ a Riemannian submanifold of $\Rn$, that is, \[\inner{v_1}{v_2}_x = v_1^\top v_2^{}.\] For that specific metric, the covariant derivative $\Ddt$ takes on a very simple form, namely, \[\Ddt v(t) = \Proj_{x(t)} v'(t)\] where $\Proj_{x} = I_n - xx^\top$ is the orthogonal projector to the tangent space at $x$. At $t = 0$, this reveals \[w = \Ddt v(0) = \Proj_{x(0)} v'(0) = \Proj_x \dot v = \dot v - (x^\top \dot v) x.\]

Say now that we have two tangent vectors $(\dot x_1, \dot v_1)$ and $(\dot x_2, \dot v_2)$ to $\T\calM$ at $(x, v)$, and we map them to two pairs $(u_1, w_1)$ and $(u_2, w_2)$. We want a Riemannian metric on $\T\calM$, and we already have a Riemannian metric on $\calM$. What to do? Well, a fairly natural option is to just take inner products of the $u$s and $w$s separately and add them up, like so: \[\begin{align} \inner{(\dot x_1, \dot v_1)}{(\dot x_2, \dot v_2)}_{(x, v)} := \inner{u_1}{u_2}_x + \inner{w_1}{w_2}_x. \end{align}\] As it turns out, this is exactly the Sasaki metric: see (Musso and Tricerri 1988, eq. (1.1)).

Continuing with the sphere example, we find that the Sasaki metric is simply \[\begin{align*} \inner{(\dot x_1, \dot v_1)}{(\dot x_2, \dot v_2)}_{(x, v)} & = \inner{\dot x_1}{\dot x_2}_x + \inner{\Proj_x \dot v_1}{\Proj_x \dot v_2}_x \\ & = \dot x_1^\top \dot x_2^{} + \dot v_1^\top \dot v_2^{} - (x^\top \dot v_1)(x^\top\dot v_2). \end{align*}\] In particular, this is not the same as the metric we would get from looking at $\T\Sn$ as a Riemannian submanifold of $\Rn \times \Rn$ (where the last term would not occur).

Beyond the sphere example, things particularize nicely when $\calM$ has the Riemannian submanifold structure inherited from the Euclidean space $\calE$. Indeed, it is then true in general that \[ \Ddt v(t) = \Proj_{x(t)} v'(t) \] with $\Proj_x$ the orthogonal projector (in the Euclidean metric) from $\calE$ to $\T_x \calM$.

This fact and all the other ones about submanifolds and Riemannian submanifolds of a Euclidean space as used in this post can be found via the table of contents in my book, with the same notation.

Thus, if $\calM$ is a Riemannian submanifold of $\calE$, it holds that \[\begin{align*} u = \dot x && \textrm{ and } && w := \Proj_x \dot v \end{align*}\] and so the Sasaki metric is simply \[\begin{align} \inner{(\dot x_1, \dot v_1)}{(\dot x_2, \dot v_2)}_{(x, v)} & = \inner{\dot x_1}{\dot x_2}_x + \inner{\Proj_x \dot v_1}{\Proj_x \dot v_2}_x \\ & = \inner{\dot x_1}{\dot x_2}_\calE + \inner{\dot v_1}{\Proj_x \dot v_2}_\calE, \end{align}\] where we also used the fact that $\Proj_x$ is self-adjoint and idempotent with respect to the Euclidean metric $\inner{\cdot}{\cdot}_\calE$.

Getting there heuristically

Let’s first aim for a more direct description of the tangent spaces to the tangent bundle.

As before, $\calM$ is a Riemannian submanifold of $\calE$, so that it inherits its metric from the Euclidean one. The tangent bundle of $\calM$ can be described using $\Proj_x$ (the orthogonal projector from $\calE$ to $\T_x\calM$), as follows: \[\begin{align*} \T\calM & = \{ (x, v) \in \calE \times \calE : x \in \calM \textrm{ and } v \in \T_x \calM \} \\ & = \{ (x, v) \in \calE \times \calE : x \in \calM \textrm{ and } \Proj_x v = v \}. \end{align*}\] Let $t \mapsto c(t) = (x(t), v(t))$ be a smooth curve on $\T\calM$ passing through $(x, v)$ at $t = 0$. Then $c'(0)$ is a tangent vector to $\T\calM$ at $(x, v)$, and every tangent vector is of that form.

We know two things:

$x(t)$ is on $\calM$ for all $t$, and also
$\Proj_{x(t)}(v(t)) = v(t)$ for all $t$.

As before, differentiating $x(t)$ reveals (by definition) that $x'(0)$ is a tangent vector to $\calM$ at $x = x(0)$. Call that $\dot x = x'(0)$.

Now differentiating the second expression, we get \[ \left(\ddt \Proj_{x(t)}\right)(v(t)) + \Proj_{x(t)}(v'(t)) = v'(t). \qquad(3)\] What is the differential of the orthogonal projector? This is where we use the fact that $\calM$ is a Riemannian submanifold of $\calE$. Indeed, we can then relate that differential to the second fundamental form $\II$: see (Boumal 2023, sec. 5.11).

Specifically, evaluating the expression above at $t = 0$ yields \[ \II(x'(0), v(0)) + \Proj_{x(0)}(v'(0)) = v'(0), \] where $\II$ is a bilinear map which sends two tangent vectors of $\calM$ at some point $x$ to a normal vector at $x$. That is, $\II(\dot x, v)$ is a vector in $\calE$ that is orthogonal to the tangent space $\T_x\calM$ with respect to the Euclidean metric.

Let $\dot v = v'(0)$ (the usual derivative of $v(t)$ at $t = 0$, seen as a curve in $\calE$). The expression above provides us with a splitting of $\dot v$ into a tangent part and a normal part: \[ \dot v = \Proj_x(\dot v) + \II(\dot x, v). \] We can also state this as follows: \[ \dot v = w + \II(\dot x, v) \textrm{ for some } w \in \T_x \calM. \] Here, we see plainly that $\dot v$ typically isn’t tangent at $x$, but its normal part is entirely determined by $\dot x$ and $v$.

To summarize, we have found that all tangent vectors $(\dot x, \dot v)$ to $\T\calM$ at $(x, v)$ are of the form \[\begin{align*} \dot x = u && \textrm{ and } && \dot v = w + \II(u, v) && \textrm{ with } && u, w \in \T_x\calM. \end{align*}\] The other way around, given $(\dot x, \dot v)$ we can easily recover $u, w$ as \[\begin{align*} u = \dot x && \textrm{ and } && w = \Proj_x(\dot v). \end{align*}\] From here, one might heuristically suggest to define the metric on $\T\calM$ by taking inner products of the $u$ parts and the $w$ parts separately, then adding them up. That leads exactly to the formula we found before.

Connecting with the axiomatic definition

The story in the previous section shows that every tangent vector to $\T_{(x, v)}\T\calM$ splits in a sum of two tangent vectors, as follows: \[\begin{align*} (\dot x, \dot v) & = (\dot x, \II(\dot x, v)) + (0, \Proj_x(\dot v)) \\ & = (u, \II(u, v)) + (0, w), \end{align*}\] with $u, w \in \T_x\calM$. As we range over all $u \in \T_x\calM$, the first vector yields a subspace of $\T_{(x, v)}\T\calM$ that has the same dimension as $\dim\calM$. It is called the horizontal space at $(x, v)$: \[ H_{(x, v)} = \{ (u, \II(u, v)) \in \calE \times \calE : u \in \T_x\calM \}. \] Likewise, collecting all the other vectors as $w$ ranges over $\T_x\calM$ yields another subspace of $\T_{(x, v)}\T\calM$, also of dimension $\dim\calM$. That one is called the vertical space at $(x, v)$: \[ V_{(x, v)} = \{ (0, w) \in \calE \times \calE : w \in \T_x\calM \}. \] Notice that if $x$ is fixed then the vertical space does not depend on $v$. That is because the vertical spaces are the tangent spaces to the fibers of the tangent bundle, which are the tangent spaces of $\calM$ (so, linear spaces).

From our expression for the Sasaki metric Eq. 2, it is straightforward to check the following:

The horizontal and vertical spaces are orthogonal to one another (and their sum is the entire tangent space). Indeed, \[\inner{(u, \II(u, v))}{(0, w)}_{(x, v)} = \inner{u}{0}_x + \inner{\Proj_x(\II(u, v))}{w}_x = 0\] for all $u, w \in \T_x\calM$, because $\II(u, v)$ is normal at $x$.
The tangent bundle projection map \[\pi \colon \T\calM \to \calM \colon (x, v) \mapsto x\] is a Riemannian submersion. Indeed, \[\D\pi(x, v) \colon \T_{(x, v)}\T\calM \to \T_{\pi(x, v)}\calM = \T_x\calM\] is simply $\D\pi(x, v)[\dot x, \dot v] = \dot x$. Its kernel is exactly the vertical space $V_x$, and the restriction of $\D\pi(x, v)$ to the horizontal space $H_{(x, v)}$ (the orthogonal complement of $V_{(x, v)}$) is an isometry from $H_{(x, v)}$ to $\T_x\calM$, that is, \[\begin{multline*}\inner{\D\pi(x, v)[u_1, \II(u_1, v)]}{\D\pi(x, v)[u_2, \II(u_2, v)]}_x \\ = \inner{u_1}{u_2}_x \\ = \inner{(u_1, \II(u_1, v))}{(u_2, \II(u_2, v))}_{(x, v)} \end{multline*}\] for all $u_1, u_2 \in \T_x\calM$. In words: given two horizontal vectors at $(x, v)$, we can take their inner product in the Sasaki metric, or push them through the differential of $\pi$ and take the inner product in the metric of $\calM$, and we’ll get the same number either way.
The Sasaki metric restricted to a fiber of $\pi$, namely, $\pi^{-1}(x) = \{(x, v) : v \in \T_x\calM\}$ reduces to the metric $\inner{\cdot}{\cdot}_x$ on $\T_x\calM$ (since $x$ is fixed and we can discard the zero component). Indeed, the tangent spaces to the fiber are the vertical spaces, and we have \[\inner{(0, w_1)}{(0, w_2)}_{(x, v)} = \inner{w_1}{w_2}_x\] for all $w_1, w_2 \in \T_x\calM$. This is indeed a Euclidean metric upon fixing $x$, because it is independent of $v$.

Those are (out of order) the three defining properties of the Sasaki metric.

References

Boumal, N. 2023. An Introduction to Optimization on Smooth Manifolds. Cambridge University Press. https://doi.org/10.1017/9781009166164.

Musso, E., and F. Tricerri. 1988. “Riemannian Metrics on Tangent Bundles.” Annali Di Matematica Pura Ed Applicata 150 (1): 1–19. https://doi.org/10.1007/bf01761461.

Sasaki, S. 1958. “On the Differential Geometry of Tangent Bundle of Riemannian Manifolds.” Tôhoku Mathematical Journal 10: 338–54.

Citation

BibTeX citation:

@online{boumal2025,
  author = {Boumal, Nicolas},
  title = {The {Sasaki} Metric for {Riemannian} Submanifolds of
    {Euclidean} Space},
  date = {2025-08-19},
  url = {www.racetothebottom.xyz/posts/Sasaki-submanifold/},
  langid = {en},
  abstract = {Given a Riemannian manifold \$\textbackslash calM\$, the
    Sasaki metric is a natural Riemannian metric for the tangent bundle
    \$\textbackslash T\textbackslash calM\$. It is often described
    axiomatically, but it also admits a simple, transparent formula for
    the common case where \$\textbackslash calM\$ is a Riemannian
    submanifold of a Euclidean space. This post provides the formula and
    illustrates it on the sphere.}
}